Polynomial Factors and Graphs Practice Questions — SAT

Question 1Question →

When the polynomial $g(x) = x^4 - 2x^3 + ax^2 - 8$ is divided by $x - 3$ , the remainder is $37$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

According to the Polynomial Remainder Theorem, dividing a polynomial

g(x)

by a linear divisor

x - c

yields a remainder equal to

g(c)

. In this problem, the divisor is

x - 3

, so we evaluate the polynomial at

x = 3

and set it equal to the given remainder of

37

. Substituting

3

for

x

in

g(x) = x^4 - 2x^3 + ax^2 - 8

yields

3^4 - 2(3)^3 + a(3)^2 - 8 = 37

. Simplifying the numerical expressions gives

81 - 54 + 9a - 8 = 37

, which simplifies to

19 + 9a = 37

. Subtracting

19

from both sides results in

9a = 18

. Dividing both sides by

9

gives the value of the constant

a

as

2

.

Step-by-Step Solution

1

Apply the Polynomial Remainder Theorem

g(3) = 37

The Remainder Theorem states that when a polynomial

g(x)

is divided by

x - c

, the remainder is

g(c)

.

2

Substitute

x = 3

into the polynomial

g(x)

3^4 - 2(3)^3 + a(3)^2 - 8 = 37

This sets the value of the polynomial evaluated at

x = 3

equal to the given remainder of

37

.

3

Simplify the constant terms

19 + 9a = 37

Evaluating the exponents and multiplying:

3^4 = 81

,

2(3^3) = 54

, and

a(3^2) = 9a

. Combining the constant terms gives

81 - 54 - 8 = 19

.

4

Solve the linear equation for

a

a = 2

Subtracting

19

from both sides gives

9a = 18

, and dividing both sides by

9

gives

a = 2

.

Key Concept

Polynomial Remainder Theorem

Question 2Question →

If a polynomial $p(x)$ is divided by $x - 5$ , the remainder is $12$ . Which of the following equations must be true?

p(5) = 12

p(-5) = 12

p(5) = -12

p(12) = 5

Show answer & explanation

Answer: $p(5) = 12$

Answer

p(5) = 12

According to the Remainder Theorem, dividing a polynomial

p(x)

by a linear expression

x - c

results in a remainder equal to

p(c)

. Here, the divisor is

x - 5

, so

c = 5

. Since the remainder is

12

, the equation that must be true is

p(5) = 12

.

Step-by-Step Solution

1

Identify the divisor and its corresponding value of

c

using the Remainder Theorem.

The divisor is

x - 5

, so we set

x - 5 = 0

, which gives

c = 5

.

The Remainder Theorem states that the remainder of a polynomial

p(x)

divided by

x - c

is

p(c)

.

2

Set

p(c)

equal to the given remainder.

Since the remainder is

12

and

c = 5

, we get

p(5) = 12

.

This directly applies the relation

p(c) = \text{remainder}

.

Key Concept

The Remainder Theorem

Question 3Question →

If the polynomial $p(x) = x^2 - kx + 12$ is divisible by $x - 3$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

7

According to the Factor Theorem, if a polynomial

p(x)

is divisible by

x - c

, then

p(c) = 0

. For the polynomial

p(x) = x^2 - kx + 12

to be divisible by

x - 3

, we must have

p(3) = 0

. Substituting

x = 3

into the polynomial gives

3^2 - k(3) + 12 = 0

. Simplifying this equation yields

9 - 3k + 12 = 0

, which combines to

21 - 3k = 0

. Solving for

k

gives

3k = 21

, which simplifies to

k = 7

.

Step-by-Step Solution

1

Apply the Factor Theorem to find the root of the polynomial.

Since

x - 3

is a factor of

p(x)

, evaluating the polynomial at

x = 3

must equal 0, so

p(3) = 0

.

The Factor Theorem states that

x - c

is a factor of a polynomial

p(x)

if and only if

p(c) = 0

.

2

Substitute

x = 3

into

p(x) = x^2 - kx + 12

and set it equal to 0.

3^2 - k(3) + 12 = 0

This sets up an algebraic equation to solve for the unknown constant

k

.

3

Simplify the equation and solve for

k

.

9 - 3k + 12 = 0 \implies 21 - 3k = 0 \implies 3k = 21 \implies k = 7

Combine like terms and isolate

k

using basic algebraic operations.

Key Concept

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

.

Question 4Question →

A polynomial $p$ is defined by $p(x) = (x - 4)(x^2 + ax + 3)$ , where $a$ is a constant. If $p(1) = -18$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Substituting

x = 1

into

p(x) = (x - 4)(x^2 + ax + 3)

gives

p(1) = (1 - 4)(1^2 + a(1) + 3) = -3(4 + a)

. Since we are given that

p(1) = -18

, we can set

-3(4 + a) = -18

. Dividing both sides of the equation by

-3

gives

4 + a = 6

. Subtracting

4

from both sides gives

a = 2

.

Step-by-Step Solution

1

Substitute

x = 1

into the expression for

p(x)

.

p(1) = (1 - 4)(1^2 + a(1) + 3) = -3(4 + a)

We are given that

p(1) = -18

, so substituting

x = 1

will allow us to set up an equation in terms of the constant

a

.

2

Set the simplified expression equal to

-18

and solve for

a

.

a = 2

Dividing both sides of

-3(4 + a) = -18

by

-3

yields

4 + a = 6

, and subtracting

4

from both sides gives the value of

a

.

Key Concept

Using given function values to solve for unknown coefficients in a factored polynomial.

Question 5Question →

The graph of the polynomial function $f$ in the $xy$ -plane is defined by $f(x) = a(x - 3)(x + 4)$ , where $a$ is a constant. If the $y$ -intercept of the graph is $(0, -24)$ , what is the value of $a$ ?

-2

\frac{1}{2}

2

-12

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. The

y

-intercept of a graph is the point where

x = 0

. For the function

f(x) = a(x - 3)(x + 4)

, substituting

x = 0

yields

f(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a

. Given that the

y

-intercept is

(0, -24)

, the value of the function at

x = 0

is

-24

. Setting these two values equal gives the equation

-12a = -24

. Dividing both sides of the equation by

-12

isolates the constant, resulting in

a = 2

.

Step-by-Step Solution

1

Identify the relationship between the

y

-intercept and the function value at

x = 0

.

Since the

y

-intercept is

(0, -24)

, it follows that

f(0) = -24

.

By definition, the

y

-intercept of a graph is the point where the graph crosses the vertical axis, which occurs at

x = 0

.

2

Substitute

x = 0

into the polynomial expression for

f(x)

.

f(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a

.

This evaluates the algebraic expression at the

y

-intercept to express the value in terms of the constant

a

.

3

Equate the evaluated expression to the known

y

-intercept value and solve for

a

.

-12a = -24 \Rightarrow a = 2

.

Setting the two expressions for

f(0)

equal allows us to solve the linear equation for the constant

a

.

Key Concept

Evaluating a polynomial at

x = 0

to relate its algebraic form to its

y

-intercept.

Question 6Question →

The polynomial function $p$ is defined by $p(x) = (x - 3)(x - 5)(x - r)$ , where $r$ is a constant. If the graph of $y = p(x)$ in the $xy$ -plane intersects the $y$ -axis at $(0, -60)$ , what is the value of $r$ ?

Show answer & explanation

Answer: 4

Answer

The value of

r

is

4

.

Evaluating the polynomial at

x = 0

yields the

y

-intercept of the graph, which is given as

-60

. Substituting

0

for

x

in the equation

p(x) = (x - 3)(x - 5)(x - r)

gives

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

. Setting this expression equal to

-60

yields

-15r = -60

. Dividing both sides of the equation by

-15

yields

r = 4

.

Step-by-Step Solution

1

Identify the meaning of the

y

-intercept.

Evaluating

p(x)

at

x = 0

must equal

-60

, so

p(0) = -60

.

The

y

-intercept of a graph occurs where the

x

-coordinate is

0

.

2

Substitute

x = 0

into the polynomial expression.

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

.

This simplifies the polynomial expression in terms of the unknown constant

r

.

3

Set the expression equal to the given

y

-intercept value and solve for

r

.

-15r = -60 \implies r = 4

.

Dividing both sides by

-15

isolates the variable

r

.

Key Concept

Using the y-intercept of a polynomial function to find an unknown root/constant in its factored form.

Question 7Question →

A polynomial $p(x)$ has a remainder of $3$ when divided by $x - 5$ . Which of the following equations must be true?

p(-5) = 3

p(5) = 0

p(5) = 3

p(3) = 0

Show answer & explanation

Answer: $p(5) = 3$

Answer

The equation

p(5) = 3

must be true.

According to the Remainder Theorem, when a polynomial

p(x)

is divided by a linear expression of the form

x - c

, the remainder is equal to

p(c)

. Here, the divisor is

x - 5

, which gives

c = 5

. The remainder is given as

3

. Therefore, evaluating the polynomial at

x = 5

must yield

3

, which is written as

p(5) = 3

.

Step-by-Step Solution

1

Identify the divisor and the remainder from the given problem statement.

The divisor is

x - 5

and the remainder is

3

.

Applying the Remainder Theorem requires identifying the value of

c

in the divisor form

x - c

and the remainder value.

2

Apply the Remainder Theorem to relate the divisor and the remainder to the polynomial function.

Since the divisor is

x - 5

, we have

c = 5

. The Remainder Theorem states that the remainder when

p(x)

is divided by

x - c

is

p(c)

. Therefore,

p(5) = 3

.

This establishes the mathematical relationship directly showing which equation must be true.

Key Concept

The Remainder Theorem

Estimated Time:45s

Question 8Question →

In the polynomial function $p(x) = x^3 - 5x^2 + 2x + k$ , the constant $k$ is chosen such that $p(x)$ is divisible by $x - 4$ . What is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of

k

is 8.

According to the Factor Theorem, a polynomial

p(x)

is divisible by

x - c

if and only if

p(c) = 0

. Since

p(x)

is divisible by

x - 4

, we must have

p(4) = 0

. Substituting

4

for

x

in the equation for

p(x)

yields

4^3 - 5(4)^2 + 2(4) + k = 0

. Simplifying the numerical terms gives

64 - 80 + 8 + k = 0

, which simplifies further to

-8 + k = 0

. Therefore, the value of the constant

k

is

8

.

Step-by-Step Solution

1

Apply the Factor Theorem to relate the factor

x - 4

to the value of the polynomial.

Since

x - 4

is a factor of

p(x)

,

p(4) = 0

.

By the Factor Theorem, if

x - c

is a factor of a polynomial

p(x)

, then

p(c) = 0

.

2

Substitute

x = 4

into the polynomial expression.

4^3 - 5(4)^2 + 2(4) + k = 0

We substitute

x = 4

into

p(x) = x^3 - 5x^2 + 2x + k

and set the expression to

0

.

3

Simplify the numerical expression to solve for

k

.

64 - 80 + 8 + k = 0 \implies -8 + k = 0 \implies k = 8

Evaluate powers, multiply, and solve the resulting linear equation for

k

.

Key Concept

Factor Theorem

Estimated Time:1m 0s

Question 9Question →

The polynomial function $p$ is defined by $p(x) = x^4 - 8x^3 + 20x^2 - 16x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ is tangent to the $x$ -axis at two distinct points. What is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. A quartic polynomial that is tangent to the x-axis at two distinct points must have two roots of multiplicity 2. Thus, it can be factored as p(x) = (x - r_1)^2(x - r_2)^2 = [x^2 - (r_1 + r_2)x + r_1 r_2]^2. Setting S = r_1 + r_2 and P = r_1 r_2, we expand this to p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2. Matching this with the given equation x^4 - 8x^3 + 20x^2 - 16x + c shows that -2S = -8, so S = 4. Using the quadratic term, S^2 + 2P = 20, which yields 16 + 2P = 20, so P = 2. Finally, the constant term is c = P^2 = 2^2 = 4.

Step-by-Step Solution

1

Express the polynomial using its roots and multiplicities.

p(x) = (x - r_1)^2(x - r_2)^2

Since the graph of the quartic polynomial is tangent to the x-axis at two distinct points, it must have two real roots, each with a multiplicity of 2.

2

Group the terms and introduce variables for the sum and product of the roots.

p(x) = [x^2 - Sx + P]^2, where S = r_1 + r_2 and P = r_1 r_2

This simplifies expansion by using the symmetric properties of roots.

3

Expand the squared quadratic expression.

p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2

To compare coefficients with the standard form of the given polynomial.

4

Compare the coefficients of the x^3 term.

-2S = -8, which gives S = 4

To determine the sum of the roots from the given cubic coefficient.

5

Compare the coefficients of the x^2 term using the value of S.

S^2 + 2P = 20, which becomes 16 + 2P = 20, leading to P = 2

To find the product of the roots using the quadratic coefficient.

6

Verify the linear coefficient using S and P.

-2SP = -2(4)(2) = -16

To ensure consistency with the given coefficient of x.

7

Determine the constant term c.

c = P^2 = 2^2 = 4

To find the final value of the constant c.

Key Concept

Representing a polynomial function in factored form based on the multiplicity of its roots and using coefficient matching to determine unknown constants.

Question 10Question →

A cubic polynomial function $p$ is defined by $p(x) = a(x + 3)(x - 1)(x - k)$ , where $a$ and $k$ are constants. In the $xy$ -plane, the $y$ -intercept of the graph of $y = p(x)$ is $(0, 6)$ . The graph of the shifted function $y = p(x - 2)$ passes through the point $(4, -30)$ . What is the third $x$ -intercept of the graph of $p$ ?

-1

9

\frac{1}{3}

-\frac{1}{2}

Show answer & explanation

Answer: $-1$

Answer

-1

The correct answer is

-1

because the

y

-intercept

(0, 6)

gives

3ak = 6

, which simplifies to

ak = 2

. The shift

y = p(x-2)

passing through

(4, -30)

implies

p(2) = -30

. Substituting

x=2

into the polynomial expression yields

5a(2-k) = -30

, which expands to

10a - 5ak = -30

. Substituting

ak = 2

gives

10a - 10 = -30

, solving to

a = -2

. Using

ak = 2

, we find

k = -1

. The third factor is

(x - k) = (x + 1)

, which corresponds to the third

x

-intercept at

x = -1

.

Step-by-Step Solution

1

Use the

y

-intercept of the graph of

y = p(x)

to establish a relationship between

a

and

k

.

p(0) = a(0 + 3)(0 - 1)(0 - k) = 3ak = 6 \Rightarrow ak = 2

.

The

y

-intercept is the point on the graph where

x = 0

.

2

Translate the point on the shifted graph back to the original function

p(x)

.

p(4 - 2) = p(2) = -30

.

Since the graph of

y = p(x - 2)

passes through

(4, -30)

, substituting

x = 4

yields

y = -30

.

3

Substitute

x = 2

into the expression for

p(x)

to set up the second equation.

p(2) = a(2 + 3)(2 - 1)(2 - k) = 5a(2 - k) = 10a - 5ak = -30

.

This establishes a system of equations with the relation from Step 1.

4

Solve the system of equations by substituting

ak = 2

into the second equation.

10a - 5(2) = -30 \Rightarrow 10a - 10 = -30 \Rightarrow 10a = -20 \Rightarrow a = -2

. Since

ak = 2

, we have

-2k = 2 \Rightarrow k = -1

.

Solving for the unknown constants

a

and

k

determines the specific polynomial expression.

5

Identify the third

x

-intercept from the factored form of the polynomial.

p(x) = -2(x + 3)(x - 1)(x + 1)

. The factors correspond to roots at

x = -3

,

x = 1

, and

x = -1

. The third

x

-intercept is

-1

.

The third factor

(x - k)

becomes

(x + 1)

when

k = -1

, yielding the root and intercept at

x = -1

.

Key Concept

Polynomial Factors and Graphs

Question 11Question →

The function $f$ is defined by $f(x) = (x - 4)(x - 2)(x + k)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane is $(0, 24)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

3

The

y

-intercept of the graph of

y = f(x)

is the point where

x = 0

. Since the

y

-intercept is

(0, 24)

, it follows that

f(0) = 24

. Substituting

0

for

x

in the equation for

f(x)

gives

f(0) = (0 - 4)(0 - 2)(0 + k) = (-4)(-2)(k) = 8k

. Setting this equal to the

y

-value of the intercept yields

8k = 24

. Dividing both sides of the equation by

8

gives

k = 3

.

Step-by-Step Solution

1

Use the definition of the

y

-intercept to find the value of

f(0)

f(0) = 24

The

y

-intercept of a graph is the point where the graph crosses the

y

-axis, corresponding to

x = 0

. Given the point

(0, 24)

,

f(0)

must equal

24

.

2

Evaluate the polynomial at

x = 0

in terms of

k

f(0) = 8k

Substituting

0

for

x

in

f(x) = (x - 4)(x - 2)(x + k)

gives

f(0) = (-4)(-2)(k)

, which simplifies to

8k

.

3

Set the evaluated expression equal to the

y

-intercept value and solve for

k

k = 3

Equating

8k

to

24

and dividing both sides by

8

yields

k = 3

.

Key Concept

Evaluating a factored polynomial function at

x = 0

determines its

y

-intercept. Using a given

y

-intercept allows solving for unknown coefficients or constants within the factors of the polynomial.

Question 12Question →

The polynomial function $p$ is defined by $p(x) = x^3 + 4x^2 - 7x - 10$ . If $p(2) = 0$ , which of the following expressions must be a factor of $p(x)$ ?

x + 2

x - 2

x - 1

x - 5

Show answer & explanation

Answer: $x - 2$

Answer

The expression

x - 2

must be a factor of

p(x)

.

According to the Factor Theorem, if a polynomial

p(x)

evaluates to

0

at a certain value

x = c

(meaning

p(c) = 0

), then

(x - c)

is a factor of the polynomial. Since we are given that

p(2) = 0

, the value

2

is a root of

p(x)

, which means the expression

x - 2

must be a factor of the polynomial.

Step-by-Step Solution

1

Identify the given root of the polynomial.

The problem states that

p(2) = 0

, which means that

x = 2

is a root of the polynomial

p(x)

.

A root is any value of

x

for which the polynomial evaluates to

0

.

2

Apply the Factor Theorem to write the corresponding factor.

According to the Factor Theorem, if

c

is a root of a polynomial

p(x)

, then

(x - c)

is a factor of

p(x)

. Substituting

c = 2

gives

(x - 2)

as a factor.

The Factor Theorem directly relates the roots of a polynomial to its linear factors.

Key Concept

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

.

Question 13Question →

In the $xy$ -plane, the graph of a cubic polynomial function $p$ with real coefficients has exactly two $x$ -intercepts, at $(1, 0)$ and $(4, 0)$ . If the graph of $p$ passes through the points $(0, -8)$ and $(2, 2)$ , what is the value of $p(6)$ ?

Show answer & explanation

Answer: 10

Answer

10

A cubic polynomial with real coefficients and exactly two

x

-intercepts at

(1, 0)

and

(4, 0)

must have one root of multiplicity 1 and one root of multiplicity 2. This yields two possible forms:

p(x) = a(x - 1)^2(x - 4)

or

p(x) = a(x - 1)(x - 4)^2

. Substituting the

y

-intercept

(0, -8)

into the first form gives

a = 2

, but the resulting polynomial

p(x) = 2(x - 1)^2(x - 4)

does not pass through

(2, 2)

since

p(2) = -4

. Substituting

(0, -8)

into the second form gives

a = \frac{1}{2}

, and the resulting polynomial

p(x) = \frac{1}{2}(x - 1)(x - 4)^2

correctly passes through

(2, 2)

since

p(2) = 2

. Finally, evaluating this function at

x = 6

yields

p(6) = \frac{1}{2}(6 - 1)(6 - 4)^2 = 10

.

Step-by-Step Solution

1

Determine the possible forms of the cubic polynomial based on its

x

-intercepts.

Two possible cases: Case 1:

p(x) = a(x - 1)^2(x - 4)

or Case 2:

p(x) = a(x - 1)(x - 4)^2

, where

a

is a non-zero constant.

Since the polynomial has degree 3 and exactly two distinct

x

-intercepts, one intercept must be a root of multiplicity 2 (tangent to the

x

-axis) and the other must be a root of multiplicity 1 (crossing the

x

-axis).

2

Solve for the constant

a

in Case 1 using the

y

-intercept

(0, -8)

.

a = 2

, yielding the candidate function

p(x) = 2(x - 1)^2(x - 4)

.

Substituting

x = 0

and

p(0) = -8

into the equation for Case 1 allows us to solve for

a

.

3

Test the point

(2, 2)

in the Case 1 candidate function.

p(2) = -4 \neq 2

, which means Case 1 is incorrect.

The correct function must satisfy all given points, including

(2, 2)

.

4

Solve for the constant

a

in Case 2 using the

y

-intercept

(0, -8)

.

a = \frac{1}{2}

, yielding the candidate function

p(x) = \frac{1}{2}(x - 1)(x - 4)^2

.

Substituting

x = 0

and

p(0) = -8

into the equation for Case 2 allows us to solve for

a

.

5

Test the point

(2, 2)

in the Case 2 candidate function.

p(2) = 2

, which matches the given point.

Confirming that the Case 2 function is consistent with the point

(2, 2)

.

6

Evaluate the correct function at

x = 6

.

p(6) = 10

.

Substituting

x = 6

into the verified function

p(x) = \frac{1}{2}(x - 1)(x - 4)^2

to find the final answer.

Key Concept

Identifying the algebraic form of a polynomial from its

x

-intercepts and multiplicities, and determining unknown coefficients using coordinate points.

Question 14Question →

A polynomial function $p$ of degree 3 has $x$ -intercepts at $(-2, 0)$ with multiplicity 2, and $(3, 0)$ with multiplicity 1. In the $xy$ -plane, the graph of $y = p(x)$ intersects the $y$ -axis at $(0, 24)$ . What is the remainder when $p(x)$ is divided by $x - 1$ ?

8

16

36

-36

Show answer & explanation

Answer: $36$

Answer

The remainder when the polynomial function is divided by

x - 1

is 36.

The correct answer is 36. A polynomial with a root at

x = -2

of multiplicity 2 and a root at

x = 3

of multiplicity 1 has the form

p(x) = a(x + 2)^2(x - 3)

. Since the

y

-intercept is

(0, 24)

, we solve

p(0) = a(2)^2(-3) = 24

to find

a = -2

. Thus, the polynomial is

p(x) = -2(x + 2)^2(x - 3)

. According to the Remainder Theorem, dividing

p(x)

by

x - 1

leaves a remainder of

p(1)

. Substituting

x = 1

yields

p(1) = -2(3)^2(-2) = 36

.

Step-by-Step Solution

1

Write the general form of the cubic polynomial using its roots and multiplicities.

p(x) = a(x + 2)^2(x - 3)

Since there is an

x

-intercept at

x = -2

with multiplicity 2,

(x + 2)^2

is a factor. Since there is an

x

-intercept at

x = 3

with multiplicity 1,

(x - 3)

is a factor. Here,

a

is a constant coefficient.

2

Determine the value of the constant coefficient

a

using the

y

-intercept.

a = -2

, so

p(x) = -2(x + 2)^2(x - 3)

The graph intersects the

y

-axis at

(0, 24)

, meaning

p(0) = 24

. Substituting

x = 0

gives

p(0) = a(0 + 2)^2(0 - 3) = -12a

. Setting

-12a = 24

yields

a = -2

.

3

Apply the Remainder Theorem to find the required remainder.

The remainder is equal to

p(1)

.

By the Remainder Theorem, the remainder when a polynomial

p(x)

is divided by

x - c

is

p(c)

. Here, the divisor is

x - 1

, so we evaluate the polynomial at

x = 1

.

4

Calculate the value of

p(1)

.

p(1) = 36

Substituting

x = 1

into

p(x) = -2(x + 2)^2(x - 3)

gives

p(1) = -2(1 + 2)^2(1 - 3) = -2(9)(-2) = 36

.

Key Concept

Identifying a polynomial from its roots and multiplicities, solving for its leading coefficient using a given point, and applying the Remainder Theorem.

Question 15Question →

The function $f$ is defined by $f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ is tangent to the $x$ -axis at the point $(3, 0)$ . What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

3

For the graph of a polynomial function to be tangent to the

x

-axis at

(3, 0)

, the root

x = 3

must have an even multiplicity (at least 2). The function is defined as

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

. Since there is already one factor of

(x - 3)

explicitly defined, the remaining cubic factor

g(x) = x^3 - kx^2 + 5x - 15

must also have a factor of

(x - 3)

to make the total multiplicity of the root

x = 3

at least 2. According to the Factor Theorem, if

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

. Substituting

x = 3

into

g(x)

gives

3^3 - k(3)^2 + 5(3) - 15 = 0

, which simplifies to

27 - 9k = 0

. Solving this equation for

k

yields

k = 3

.

Step-by-Step Solution

1

Identify the relationship between graph tangency and factor multiplicity.

For the graph of a polynomial function to be tangent to the

x

-axis at a point

(c, 0)

, the factor

(x - c)

must have an even multiplicity of at least 2 in the polynomial's factored form.

An odd multiplicity root causes the graph to cross the

x

-axis, while an even multiplicity root causes the graph to touch the

x

-axis and turn around (tangency).

2

Apply the multiplicity requirement to the given function.

Since

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

already contains one factor of

(x - 3)

, the cubic expression

g(x) = x^3 - kx^2 + 5x - 15

must also contain

(x - 3)

as a factor to ensure the total multiplicity of the root

x = 3

is at least 2.

This guarantees that

(x - 3)^2

is a factor of

f(x)

.

3

Apply the Factor Theorem to the cubic expression.

Since

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

.

The Factor Theorem states that a polynomial

P(x)

has a factor

(x - c)

if and only if

P(c) = 0

.

4

Solve for the constant

k

by substituting

x = 3

into

g(x)

.

3^3 - k(3)^2 + 5(3) - 15 = 0 \implies 27 - 9k + 15 - 15 = 0 \implies 27 - 9k = 0 \implies 9k = 27 \implies k = 3

.

Arithmetic simplification yields the value of the constant.

Key Concept

The relationship between polynomial factors, root multiplicities, and the behavior of the graph at

x

-intercepts.

Question 16Question →

A polynomial function $q$ with real coefficients satisfies the equation $q(x) + q(6-x) = 8$ for all real numbers $x$ . In the $xy$ -plane, the graph of $y = q(x)$ has an $x$ -intercept at $(5, 0)$ . What is the remainder when $q(x)$ is divided by $x-1$ ?

8

-8

0

2

Show answer & explanation

Answer: $8$

Answer

The correct answer is

8

, which is the remainder when the polynomial is divided by

x-1

.

To find the remainder when the polynomial

q(x)

is divided by

x-1

, we use the Remainder Theorem, which states that this remainder is equal to

q(1)

. Since the graph of

y = q(x)

has an

x

-intercept at

(5, 0)

, we know that

q(5) = 0

. Substituting

x = 5

into the given equation

q(x) + q(6-x) = 8

yields

q(5) + q(6-5) = 8

, which simplifies to

q(5) + q(1) = 8

. Substituting

q(5) = 0

into this equation gives

0 + q(1) = 8

, so

q(1) = 8

. Geometrically, the equation

\frac{q(x) + q(6-x)}{2} = 4

shows that the graph of

q

is symmetric under a point reflection about

(3, 4)

. Reflecting the point

(5, 0)

about

(3, 4)

gives the point

(1, 8)

, verifying that

q(1) = 8

.

Step-by-Step Solution

1

Translate the given graphical intercept into function notation.

q(5) = 0

An

x

-intercept at

(5, 0)

means that when the input

x

is

5

, the output of the function is

0

.

2

Apply the Remainder Theorem to the divisor

x-1

.

The remainder is equal to

q(1)

.

The Remainder Theorem states that when a polynomial

q(x)

is divided by

x-a

, the remainder is

q(a)

.

3

Substitute

x = 5

into the given functional equation

q(x) + q(6-x) = 8

.

q(5) + q(6-5) = 8 \implies q(5) + q(1) = 8

Substituting

x = 5

allows us to relate the known value

q(5)

directly to the target value

q(1)

.

4

Solve for

q(1)

using the value of

q(5)

.

0 + q(1) = 8 \implies q(1) = 8

Substituting

q(5) = 0

into the equation yields the value of

q(1)

, which is the remainder.

Key Concept

Remainder Theorem and Polynomial Graphs

Alternative Method

Use the point symmetry of the graph. The equation

q(x) + q(6-x) = 8

can be rewritten as

\frac{q(x) + q(6-x)}{2} = 4

. This indicates that the graph of

y = q(x)

has point symmetry about the point

(3, 4)

. Since the graph has an

x

-intercept at

(5, 0)

, point symmetry dictates that the reflected point across

(3, 4)

must also lie on the graph. The midpoint of

(5, 0)

and the symmetric point

(x_0, y_0)

is

(3, 4)

. Solving

\frac{5+x_0}{2} = 3

gives

x_0 = 1

, and

\frac{0+y_0}{2} = 4

gives

y_0 = 8

. Thus,

(1, 8)

lies on the graph, meaning

q(1) = 8

. By the Remainder Theorem, the remainder when

q(x)

is divided by

x-1

is

q(1) = 8

.

Estimated Time:1m 30s

Question 17Question →

The polynomial function $p$ is defined by $p(x) = x^3 + bx^2 + cx + d$ , where $b$ , $c$ , and $d$ are constants. In the $xy$ -plane, the graph of $y = p(x)$ has $x$ -intercepts at $(2, 0)$ and $(-3, 0)$ . If the remainder when $p(x)$ is divided by $x - 1$ is $-8$ , what is the value of $d$ ?

-6

6

-2

18

Show answer & explanation

Answer: $-6$

Answer

-6

The correct value is

-6

. Since the graph has

x

-intercepts at

(2, 0)

and

(-3, 0)

, the Factor Theorem dictates that

(x - 2)

and

(x + 3)

are factors of

p(x)

. Since the leading coefficient of the cubic polynomial

p(x)

is

1

, it can be written in factored form as

p(x) = (x - 2)(x + 3)(x - r)

, where

r

is the third root. According to the Remainder Theorem, the remainder when

p(x)

is divided by

x - 1

is equal to

p(1) = -8

. Substituting

x = 1

yields

-4(1 - r) = -8

, which simplifies to

1 - r = 2

, so

r = -1

. The polynomial is thus

p(x) = (x - 2)(x + 3)(x + 1)

. The constant term

d

is equal to the value of the function when

x = 0

, which is

p(0) = (-2)(3)(1) = -6

.

Step-by-Step Solution

1

Determine the factors corresponding to the given

x

-intercepts of the polynomial.

The factors are

(x - 2)

and

(x + 3)

.

Since the graph of

y = p(x)

has

x

-intercepts at

(2, 0)

and

(-3, 0)

, we know that

p(2) = 0

and

p(-3) = 0

. According to the Factor Theorem,

(x - 2)

and

(x + 3)

must be factors of

p(x)

.

2

Set up the factored form of the cubic polynomial function

p(x)

.

p(x) = (x - 2)(x + 3)(x - r)

Because

p(x)

is a cubic polynomial (degree 3) with a leading coefficient of

1

, it can be written as the product of three linear factors:

(x - 2)

,

(x + 3)

, and

(x - r)

, where

r

is the unknown third root.

3

Apply the Remainder Theorem to solve for the third root

r

.

r = -1

By the Remainder Theorem, the remainder when

p(x)

is divided by

x - 1

is equal to

p(1)

. We are given that this remainder is

-8

, so

p(1) = -8

. Substituting

x = 1

into our factored expression gives:

p(1) = (1 - 2)(1 + 3)(1 - r) = -8 \Rightarrow -1 \cdot 4 \cdot (1 - r) = -8 \Rightarrow -4(1 - r) = -8 \Rightarrow 1 - r = 2 \Rightarrow r = -1

.

4

Determine the value of the constant coefficient

d

.

d = -6

Using the root

r = -1

, the complete factored expression is

p(x) = (x - 2)(x + 3)(x + 1)

. The constant term

d

is equivalent to

p(0)

:

d = p(0) = (0 - 2)(0 + 3)(0 + 1) = (-2)(3)(1) = -6

.

Key Concept

Using the Factor Theorem and Remainder Theorem to find unknown coefficients in a polynomial function.

Alternative Method

Alternatively, you can expand the general form

p(x) = (x - 2)(x + 3)(x - r) = (x^2 + x - 6)(x - r) = x^3 + (1 - r)x^2 - (r + 6)x + 6r

. Comparing this to

p(x) = x^3 + bx^2 + cx + d

, we see that

d = 6r

. Since the remainder when

p(x)

is divided by

x - 1

is

-8

, we have

p(1) = -8

. Substituting

x = 1

into our expanded form gives

1^3 + (1 - r)(1)^2 - (r + 6)(1) + 6r = -8 \Rightarrow 1 + 1 - r - r - 6 + 6r = -8 \Rightarrow 4r - 4 = -8 \Rightarrow 4r = -4 \Rightarrow r = -1

. Thus,

d = 6(-1) = -6

.

Estimated Time:2m 0s

Question 18Question →

A polynomial function $P(x)$ of degree 4 with real coefficients is symmetric about the line $x = 2$ in the $xy$ -plane. If $P(x)$ is divisible by $x^2 - 4x + 3$ , the remainder when $P(x)$ is divided by $x - 4$ is $36$ , and $P(2) = -8$ , what is the value of $P(5)$ ?

Show answer & explanation

Answer: 136

Answer

The value of

P(5)

is 136.

The correct answer of 136 is found by representing the symmetric fourth-degree polynomial as

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

, solving for the coefficients using the roots at

x=3

, the point

(2, -8)

, and the remainder point

(4, 36)

, and then evaluating the resulting expression

(x-2)^4 + 7(x-2)^2 - 8

at

x=5

.

Step-by-Step Solution

1

Express the fourth-degree polynomial in a form that reflects its symmetry.

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

Since the graph of

P(x)

is symmetric about the line

x = 2

, the polynomial expression in terms of

(x - 2)

must contain only even powers.

2

Use the divisibility condition to establish an equation for the coefficients.

P(3) = a(3-2)^4 + b(3-2)^2 + c = a + b + c = 0

The divisor

x^2 - 4x + 3

factors into

(x-1)(x-3)

. By the Factor Theorem,

P(3) = 0

and

P(1) = 0

.

3

Use the given value

P(2) = -8

to find the constant term

c

.

c = -8

Substituting

x = 2

into

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

makes the terms with

(x-2)

equal to zero, leaving

P(2) = c

.

4

Formulate a system of linear equations for

a

and

b

.

a + b = 8

and

4a + b = 11

Substituting

c = -8

into

a + b + c = 0

gives

a + b = 8

. By the Remainder Theorem,

P(4) = 36

, which gives

a(4-2)^4 + b(4-2)^2 - 8 = 36 \implies 16a + 4b = 44 \implies 4a + b = 11

.

5

Solve the system of equations for

a

and

b

.

a = 1

and

b = 7

Subtracting

a + b = 8

from

4a + b = 11

yields

3a = 3 \implies a = 1

, which then gives

b = 7

.

6

Evaluate the polynomial at

x = 5

.

P(5) = 136

Substitute

a = 1

,

b = 7

,

c = -8

, and

x = 5

into the symmetric polynomial form:

P(5) = (5-2)^4 + 7(5-2)^2 - 8 = 3^4 + 7(3^2) - 8 = 81 + 63 - 8 = 136

.

Key Concept

Polynomial Factors and Graphs

Question 19Question →

The function $f$ is defined by $f(x) = x^3 - 3x^2 - 10x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has $x$ -intercepts at $(c, 0)$ and $(2c, 0)$ , where $c$ is a positive constant. What is the value of $k$ ?

Show answer & explanation

Answer: 24

Answer

24

By applying the factor theorem to the two

x

-intercepts

(c,0)

and

(2c,0)

, we establish the system

c^3 - 3c^2 - 10c + k = 0

and

8c^3 - 12c^2 - 20c + k = 0

. Subtracting these equations yields

7c^3 - 9c^2 - 10c = 0

. Since

c > 0

, dividing by

c

gives

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2)=0

, yielding the positive solution

c=2

. Substituting

c=2

back into the first equation gives

8 - 12 - 20 + k = 0

, which solves to

k = 24

.

Step-by-Step Solution

1

Set up equations for the roots

c

and

2c

using the factor theorem.

f(c) = c^3 - 3c^2 - 10c + k = 0

and

f(2c) = 8c^3 - 12c^2 - 20c + k = 0

An

x

-intercept at

(r, 0)

means that

r

is a root of the polynomial, so

f(r) = 0

.

2

Subtract the first equation from the second to eliminate the constant

k

.

7c^3 - 9c^2 - 10c = 0

Eliminating

k

allows us to solve for the root

c

directly.

3

Solve the polynomial equation for the positive constant

c

.

c = 2

Dividing the equation by

c

(since

c > 0

) yields

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2) = 0

. Since

c

must be positive,

c = 2

.

4

Substitute

c = 2

back into the equation for

f(c) = 0

to solve for

k

.

k = 24

Substituting the known root value allows us to find the value of the constant coefficient

k

.

Key Concept

The relationship between a polynomial's algebraic factors, its roots, and its

x

-intercepts in the coordinate plane.

Estimated Time:2m 0s

Question 20Question →

In the $xy$ -plane, the graph of the cubic function $f(x) = x^3 - 7x^2 + kx - 12$ , where $k$ is a constant, is tangent to the $x$ -axis at one point and intersects the $x$ -axis at another point. If all roots of $f(x)$ are real numbers, which of the following could be the value of $k$ ?

16

-16

7

12

Show answer & explanation

Answer: 16

Answer

16

The correct answer is the value that makes the cubic function have a double root at

2

and a single root at

3

. Setting

f(x) = (x - 2)^2(x - 3)

and expanding it yields

x^3 - 7x^2 + 16x - 12

. Comparing this to

f(x) = x^3 - 7x^2 + kx - 12

shows that

k = 16

.

Step-by-Step Solution

1

Express the cubic function in factored form based on its roots.

f(x) = (x - r_1)^2(x - r_2)

Since the graph is tangent to the

x

-axis at one point and intersects it at another, it has one real root of multiplicity 2 (let it be

r_1

) and one real root of multiplicity 1 (let it be

r_2

).

2

Expand the factored form to obtain coefficients in terms of

r_1

and

r_2

.

f(x) = x^3 - (2r_1 + r_2)x^2 + (r_1^2 + 2r_1 r_2)x - r_1^2 r_2

Comparing this expansion to the given expression

f(x) = x^3 - 7x^2 + kx - 12

allows us to set up a system of equations.

3

Set up and solve the system of equations for

r_1

and

r_2

.

2r_1 + r_2 = 7

,

r_1^2 r_2 = 12

, and

k = r_1^2 + 2r_1 r_2

. Substituting

r_2 = 7 - 2r_1

into the second equation gives

2r_1^3 - 7r_1^2 + 12 = 0

. Testing integer root candidates shows

r_1 = 2

is a solution, which yields

r_2 = 3

.

This determines the values of the roots

r_1

and

r_2

.

4

Calculate the value of

k

using the values of the roots.

k = 2^2 + 2(2)(3) = 16

Substituting

r_1 = 2

and

r_2 = 3

into the expression for

k

gives the target coefficient.

Key Concept

Analyzing the relationship between a polynomial's algebraic factors, roots, and its graphical features such as tangency and intercepts.

Polynomial Factors and Graphs

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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