All practice questions

Question 1781Question →

If the expression $x^4 - 8x^2 + 16 - 9y^2$ is factored completely over the integers, the product of the factors can be written as $(x^2 - a - by)(x^2 - c + dy)$ , where $a$ , $b$ , $c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 14

Answer

14

By grouping the first three terms, the expression

x^4 - 8x^2 + 16

is recognized as

(x^2 - 4)^2

. Substituting this back into the original expression gives

(x^2 - 4)^2 - (3y)^2

. Applying the difference of squares identity, this factors into

(x^2 - 4 - 3y)(x^2 - 4 + 3y)

. Comparing this result to

(x^2 - a - by)(x^2 - c + dy)

where

a, b, c, d

are positive integers yields

a = 4

,

b = 3

,

c = 4

, and

d = 3

. Summing these values gives

4 + 3 + 4 + 3 = 14

.

Step-by-Step Solution

1

Group the first three terms of the polynomial.

x^4 - 8x^2 + 16 = (x^2 - 4)^2

To recognize the perfect square trinomial structure in terms of

x^2

.

2

Rewrite the original expression using the grouped terms.

(x^2 - 4)^2 - 9y^2 = (x^2 - 4)^2 - (3y)^2

To express the polynomial as a difference of squares.

3

Factor the expression using the difference of squares formula

A^2 - B^2 = (A - B)(A + B)

.

(x^2 - 4 - 3y)(x^2 - 4 + 3y)

To obtain the completely factored form over the integers.

4

Compare the factored expression to the given template

(x^2 - a - by)(x^2 - c + dy)

where

a, b, c, d

are positive integers.

a = 4

,

b = 3

,

c = 4

,

d = 3

To identify the values of the constants that satisfy the positivity constraint.

5

Calculate the sum of the identified values.

a + b + c + d = 4 + 3 + 4 + 3 = 14

To answer the question.

Key Concept

Factoring by grouping and the difference of squares

Question 1782Question →

What is the sum of all real values of

x

that satisfy the equation

\frac{x^2 - 3x}{x^2 - 3x + 2} + \frac{x^2 - 3x - 1}{x^2 - 3x - 3} = \frac{11}{3}

?

Show answer & explanation

Answer: 3

Answer

The sum of all real values of

x

that satisfy the equation is

3

.

Substituting

y = x^2 - 3x

simplifies the original rational equation into the quadratic form

y^2 - y - 12 = 0

. Solving for

y

yields the values

4

and

-3

. Substituting back

x^2 - 3x

for

y

produces two quadratic equations. The first,

x^2 - 3x - 4 = 0

, has real solutions of

4

and

-1

. The second,

x^2 - 3x + 3 = 0

, has a negative discriminant and produces no real solutions. Summing the valid real solutions gives

4 + (-1) = 3

.

Step-by-Step Solution

1

Substitute

y = x^2 - 3x

into the equation to simplify the rational terms.

\frac{y}{y+2} + \frac{y-1}{y-3} = \frac{11}{3}

Using a temporary variable simplifies the algebraic manipulation of the rational expressions.

2

Multiply the entire equation by the least common denominator,

3(y+2)(y-3)

, to eliminate all fractions.

3y(y-3) + 3(y-1)(y+2) = 11(y+2)(y-3)

, where

y \neq -2

and

y \neq 3

This clears the denominators so the equation can be solved as a polynomial.

3

Expand the terms on both sides of the equation.

(3y^2 - 9y) + (3y^2 + 3y - 6) = 11y^2 - 11y - 66

Expanding allows for combining like terms.

4

Combine like terms and move all terms to one side of the equation to set it equal to zero.

5y^2 - 5y - 60 = 0

This sets up the expression in the standard quadratic form

ay^2 + by + c = 0

.

5

Divide the entire quadratic equation by its greatest common factor,

5

, and factor the resulting expression.

y^2 - y - 12 = 0 \implies (y-4)(y+3) = 0 \implies y = 4 \text{ or } y = -3

Factoring solves for the possible values of the substituted variable

y

.

6

Substitute back

y = x^2 - 3x

for each case and solve the resulting quadratic equations for

x

.

For

y = 4

:

x^2 - 3x - 4 = 0 \implies (x-4)(x+1) = 0 \implies x = 4 \text{ or } x = -1

.
For

y = -3

:

x^2 - 3x + 3 = 0

. The discriminant is

(-3)^2 - 4(1)(3) = -3 < 0

, which means there are no real solutions.

This determines the real values of

x

that solve the original equation.

7

Verify that neither solution makes the original denominators zero, and add the valid real solutions.

4 + (-1) = 3

Since both

x = 4

and

x = -1

result in non-zero denominators, both are valid real solutions. Their sum is

3

.

Key Concept

Solving rational equations by utilizing algebraic substitution to reduce complexity and analyzing quadratic equations for real solutions.

Question 1783Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by:

z = \frac{1 - 3i^{45}}{1 + i^{95}}

Which of the following is equivalent to $z^2$ ?

-3 + 4i

5 - 4i

3 - 4i

3

5

Show answer & explanation

Answer: 3 - 4i

Answer

3 - 4i

The correct answer is found by first simplifying the powers of

i

in the expression for

z

:

i^{45} = i

and

i^{95} = -i

, which yields

z = \frac{1 - 3i}{1 - i}

. Rationalizing the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator (

1 + i

) simplifies the expression to

z = 2 - i

. Finally, squaring this result using the binomial expansion formula gives

(2 - i)^2 = 4 - 4i + i^2 = 3 - 4i

.

Step-by-Step Solution

1

Simplify the high integer powers of the imaginary unit

i

by using the fact that powers of

i

repeat in a cycle of four:

i^1 = i

,

i^2 = -1

,

i^3 = -i

, and

i^4 = 1

.

i^{45} = i

and

i^{95} = -i

Since

45 = 4(11) + 1

, the remainder is

1

, so

i^{45} = i^1 = i

. Since

95 = 4(23) + 3

, the remainder is

3

, so

i^{95} = i^3 = -i

.

2

Substitute these simplified values back into the expression for

z

.

z = \frac{1 - 3i}{1 - i}

This sets up the fraction with simplified imaginary terms in both the numerator and the denominator.

3

Rationalize the denominator by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

1 + i

.

z = 2 - i

Multiplying by the conjugate eliminates the imaginary unit from the denominator:

\frac{(1 - 3i)(1 + i)}{(1 - i)(1 + i)} = \frac{1 + i - 3i - 3i^2}{1 - i^2} = \frac{4 - 2i}{2} = 2 - i

.

4

Calculate the value of

z^2

by squaring the simplified complex number

2 - i

.

z^2 = 3 - 4i

Using the binomial squaring formula

(a - b)^2 = a^2 - 2ab + b^2

, we expand

(2 - i)^2

to get

2^2 - 2(2)(i) + i^2 = 4 - 4i - 1 = 3 - 4i

.

Key Concept

Simplifying complex numbers by evaluating powers of

i

, rationalizing fractions with complex conjugates, and expanding complex binomials.

Estimated Time:1m 30s

Question 1784Question →

When the expression $-(p - 2q)^3 - p(p - 3q)(2p + q) + 3q^2(p - q)$ is completely simplified by combining like terms, what is the coefficient of $p^2q$ ?

Show answer & explanation

Answer: 11

Answer

The coefficient of

p^2q

in the fully simplified expression is 11.

Expanding the three components of the expression yields:

-(p - 2q)^3 = -p^3 + 6p^2q - 12pq^2 + 8q^3

;

-p(p - 3q)(2p + q) = -2p^3 + 5p^2q + 3pq^2

; and

3q^2(p - q) = 3pq^2 - 3q^3

. Combining these terms gives the simplified polynomial

-3p^3 + 11p^2q - 6pq^2 + 5q^3

. The coefficient of

p^2q

is 11.

Step-by-Step Solution

1

Expand and negate the term

-(p - 2q)^3

-p^3 + 6p^2q - 12pq^2 + 8q^3

Using the binomial theorem expansion for

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

and distributing the negative sign.

2

Multiply and distribute

-p(p - 3q)(2p + q)

-2p^3 + 5p^2q + 3pq^2

First multiply the binomials

(p - 3q)(2p + q) = 2p^2 - 5pq - 3q^2

, and then multiply each term by

-p

.

3

Distribute

3q^2(p - q)

3pq^2 - 3q^3

Multiply

3q^2

by both terms inside the binomial.

4

Combine the like terms of

p^2q

11

Identify and sum all terms containing

p^2q

:

6p^2q + 5p^2q = 11p^2q

.

Key Concept

Simplifying algebraic expressions by expanding polynomials, applying the distributive property with negative signs, and combining like terms.

Question 1785Question →

Let $i$ be the imaginary unit such that $i^2 = -1$ . What is the simplified form of the expression $(1 + 2i)^2(3 - i)$ ?

-9 + 3i

-5 + 15i

19 + 7i

-13 + 15i

50i

Show answer & explanation

Answer: $-5 + 15i$

Answer

-5 + 15i

The correct answer is

-5 + 15i

. We first expand the squared binomial

(1 + 2i)^2

, which results in

1 + 4i + 4i^2

. Substituting

i^2 = -1

yields

-3 + 4i

. We then multiply this by the second binomial

(3 - i)

to get

-9 + 3i + 12i - 4i^2

. Substituting

i^2 = -1

one more time and combining like terms leads to the final simplified result of

-5 + 15i

.

Step-by-Step Solution

1

Expand the squared binomial

(1 + 2i)^2

1 + 4i + 4i^2

Before multiplying by the second binomial, we must apply the exponent to the first binomial according to the order of operations.

2

Substitute

i^2 = -1

to simplify the expression from Step 1

-3 + 4i

Since

i^2 = -1

, the term

4i^2

becomes

4(-1) = -4

, and combining the real parts gives

1 - 4 = -3

.

3

Multiply the simplified term by

(3 - i)

using the FOIL method

-9 + 3i + 12i - 4i^2

We distribute each term of the first binomial into the second binomial:

(-3)(3) = -9

,

(-3)(-i) = 3i

,

(4i)(3) = 12i

, and

(4i)(-i) = -4i^2

.

4

Simplify the resulting expression by combining like terms and substituting

i^2 = -1

-5 + 15i

Combining the imaginary parts gives

3i + 12i = 15i

. Substituting

i^2 = -1

into

-4i^2

gives

-4(-1) = +4

. Finally, combining the real parts yields

-9 + 4 = -5

.

Key Concept

Simplification of complex expressions involving binomial squaring and multiplication under the definition

i^2 = -1

.

Estimated Time:1m 30s

Question 1786Question →

If $k$ is a positive real number such that $\sqrt{k} + \sqrt{4k} = 15$ , what is the value of $k$ ?

3

5

9

25

45

Show answer & explanation

Answer: 25

Answer

25

The correct answer is

25

. To find this, we simplify

\sqrt{4k}

to

2\sqrt{k}

. Substituting this into the equation gives

\sqrt{k} + 2\sqrt{k} = 15

. Combining like terms yields

3\sqrt{k} = 15

. Dividing both sides by

3

gives

\sqrt{k} = 5

. Squaring both sides of the equation results in

k = 25

. Checking the solution:

\sqrt{25} + \sqrt{4(25)} = 5 + 10 = 15

, which is true.

Step-by-Step Solution

1

Simplify the radical term

\sqrt{4k}

using the product property of radicals.

\sqrt{4k} = \sqrt{4} \cdot \sqrt{k} = 2\sqrt{k}

This allows us to express both terms in the equation using the same radical base,

\sqrt{k}

.

2

Substitute the simplified term back into the equation and combine like terms.

\sqrt{k} + 2\sqrt{k} = 15 \implies 3\sqrt{k} = 15

Combining like radical terms simplifies the equation to a single radical term.

3

Isolate the radical by dividing both sides of the equation by

3

.

\sqrt{k} = 5

Isolating the radical term is necessary before squaring both sides to solve for the variable.

4

Square both sides of the equation to solve for

k

.

k = 25

Squaring is the inverse operation of taking the square root, which isolates

k

.

Key Concept

Solving radical equations by simplifying and combining like radical terms

Question 1787Question →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the real part of the complex number $z = (2 - i)^3 + 9i$ ?

Show answer & explanation

Answer: 2

Answer

The real part of the complex number is 2.

Expanding the expression

(2 - i)^3

yields

2 - 11i

. Adding

9i

gives

2 - 2i

. The real part of this complex number is the term without

i

, which is 2.

Step-by-Step Solution

1

Expand the squared binomial

(2 - i)^2

.

3 - 4i

Apply the identity

(a - b)^2 = a^2 - 2ab + b^2

and substitute

i^2 = -1

.

2

Multiply the result of the square by

(2 - i)

to calculate

(2 - i)^3

.

2 - 11i

Distribute the terms

(3 - 4i)(2 - i) = 6 - 3i - 8i + 4i^2

and substitute

i^2 = -1

.

3

Add

9i

to the simplified cube to find the complex number

z

.

2 - 2i

Combine the imaginary components:

-11i + 9i = -2i

.

4

Identify the real part of

z

.

2

The real part of a complex number

a + bi

is

a

.

Key Concept

Expanding complex binomials and simplifying powers of the imaginary unit

Question 1788Question →

For all real numbers $y$ that satisfy the inequality $||y - 3| - 6| \leq 2$ , what is the sum of all possible integer values of $y$ ?

-15

30

45

51

65

Show answer & explanation

Answer: $30$

Answer

The sum of all possible integer values of

y

is

30

.

The compound inequality

4 \leq |y - 3| \leq 8

splits into two parts:

|y - 3| \leq 8

(which gives

-5 \leq y \leq 11

) and

|y - 3| \geq 4

(which gives

y \leq -1

or

y \geq 7

). The intersection of these intervals is

[-5, -1] \cup [7, 11]

. Summing all the integers in these intervals gives

(-5 + -4 + -3 + -2 + -1) + (7 + 8 + 9 + 10 + 11) = -15 + 45 = 30

.

Step-by-Step Solution

1

Set up the compound inequality representing the outer absolute value.

-2 \leq |y - 3| - 6 \leq 2

By definition,

|u| \leq a

is equivalent to

-a \leq u \leq a

for

a \geq 0

.

2

Isolate the inner absolute value term by adding

6

to all parts of the inequality.

4 \leq |y - 3| \leq 8

Isolating the absolute value allows us to split the compound inequality into two separate cases.

3

Split the compound inequality into two separate inequalities and solve each one.

|y - 3| \leq 8

and

|y - 3| \geq 4

The expression

|y - 3|

must be simultaneously less than or equal to

8

and greater than or equal to

4

.

4

Solve the first inequality,

|y - 3| \leq 8

.

-5 \leq y \leq 11

Rewriting the inequality gives

-8 \leq y - 3 \leq 8

, and adding

3

to all parts yields the interval

[-5, 11]

.

5

Solve the second inequality,

|y - 3| \geq 4

.

y \leq -1

or

y \geq 7

By definition,

|u| \geq a

is equivalent to

u \geq a

or

u \leq -a

. Thus,

y - 3 \geq 4 \implies y \geq 7

, and

y - 3 \leq -4 \implies y \leq -1

.

6

Determine the intersection of the two solution sets.

y \in [-5, -1] \cup [7, 11]

The values of

y

must lie within

[-5, 11]

and also satisfy

y \leq -1

or

y \geq 7

.

7

Identify the integers within the final intervals and calculate their sum.

Sum =

30

The integers in

[-5, -1]

are

-5, -4, -3, -2, -1

(sum =

-15

). The integers in

[7, 11]

are

7, 8, 9, 10, 11

(sum =

45

). The total sum is

-15 + 45 = 30

.

Key Concept

Solving compound and nested absolute value inequalities

Alternative Method

Instead of solving the inequality algebraically, one can test the integer values around the critical points. Since

y

must satisfy

4 \leq |y - 3| \leq 8

, we can see that the distance of

y

from

3

must be between

4

and

8

units. The integers at distance

4, 5, 6, 7, 8

to the right of

3

are

7, 8, 9, 10, 11

. The integers at distance

4, 5, 6, 7, 8

to the left of

3

are

-1, -2, -3, -4, -5

. Summing these ten integers yields

30

.

Estimated Time:3m 0s

Question 1789Question →

When solving the radical equation $\sqrt{2x^2 - 4x - 6} = x - 3$ for all real values of $x$ , one of the solutions obtained from the squared equation is extraneous. What is the value of this extraneous solution?

-5

3

1

-1/2

5

Show answer & explanation

Answer: -5

Answer

The value of the extraneous solution is

-5

.

The correct answer is

-5

. Squaring both sides of the equation

\sqrt{2x^2 - 4x - 6} = x - 3

yields

2x^2 - 4x - 6 = x^2 - 6x + 9

, which simplifies to

x^2 + 2x - 15 = 0

. Factoring this quadratic gives

(x + 5)(x - 3) = 0

, leading to potential solutions of

3

and

-5

. Substituting

-5

back into the original equation results in the left side simplifying to

8

and the right side simplifying to

-8

. Since

8 \neq -8

, the value

-5

is an extraneous solution.

Step-by-Step Solution

1

Square both sides of the original radical equation to eliminate the square root.

2x^2 - 4x - 6 = (x - 3)^2

Squaring both sides removes the radical on the left side.

2

Expand the right side and move all terms to the left side to set the quadratic equation to zero.

x^2 + 2x - 15 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting this expression from both sides yields the simplified quadratic equation.

3

Factor the quadratic equation to find the potential solutions.

(x + 5)(x - 3) = 0

, which gives

x = -5

and

x = 3

.

The factors of

-15

that add up to

2

are

5

and

-3

.

4

Substitute each potential solution back into the original equation to check for extraneous roots.

For

x = 3

,

\sqrt{2(3)^2 - 4(3) - 6} = 3 - 3

simplifies to

0 = 0

(valid). For

x = -5

,

\sqrt{2(-5)^2 - 4(-5) - 6} = -5 - 3

simplifies to

8 = -8

(invalid).

Extraneous solutions satisfy the squared equation but do not satisfy the original radical equation due to the sign difference introduced by squaring.

Key Concept

Solving radical equations and identifying extraneous solutions

Estimated Time:2m 0s

Question 1790Question →

A dataset consists of $7$ positive integers sorted in non-decreasing order. The mean of the dataset is $20$ , the median is $18$ , and the unique mode is $15$ . If the range of the dataset is $22$ , what is the minimum possible value of the largest number in the dataset?

27

29

30

35

37

Show answer & explanation

Answer: 30

Answer

30

The correct answer is 30 because minimizing the largest value requires minimizing the smallest value under the sum and uniqueness constraints. When the smallest value is 8, the largest value is 30, which allows for a valid, sorted sequence of positive integers where 15 is the unique mode: {8, 15, 15, 18, 26, 28, 30}.

Step-by-Step Solution

1

Set up the variables for the sorted dataset.

Let the seven positive integers in non-decreasing order be

x_1, x_2, x_3, x_4, x_5, x_6, x_7

. Since the median is the 4th value,

x_4 = 18

. The mean is

20

, so the sum of all elements is

7 \times 20 = 140

.

Establishing standard notation and utilizing the definitions of median and mean.

2

Incorporate the range into the sum equation.

x_1 + x_2 + x_3 + 18 + x_5 + x_6 + x_7 = 140

. Since the range is

22

, the largest number is

x_7 = x_1 + 22

. Substituting this into the sum gives

2x_1 + x_2 + x_3 + x_5 + x_6 = 100

.

Simplifying the system of equations by expressing the largest element in terms of the smallest element.

3

Apply the mode constraint.

The unique mode is

15

. Since

x_4 = 18

and the dataset is sorted,

15

must be in the lower half of the dataset and must repeat. Thus,

x_2 = x_3 = 15

. The equation simplifies to

2x_1 + 30 + x_5 + x_6 = 100 \implies 2x_1 + x_5 + x_6 = 70

.

Using the properties of the mode and sorted list to lock the values of the second and third elements.

4

Set up inequalities to minimize the largest value.

To minimize

x_7 = x_1 + 22

, we must minimize

x_1

, which is equivalent to maximizing

x_5

and

x_6

. Since

15

is the unique mode with a frequency of

2

, no other value can repeat. Thus,

x_5, x_6,

and

x_7

must be distinct from each other and larger than the median

18

. This means

x_5 \ge 19

. The upper bounds are

x_6 \le x_7 - 1 = x_1 + 21

and

x_5 \le x_7 - 2 = x_1 + 20

.

Applying the uniqueness constraint of the mode to restrict the values of the upper half of the dataset.

5

Solve for the minimum integer value of the smallest element.

Substitute the maximum bounds of

x_5

and

x_6

into the equation:

2x_1 + (x_1 + 20) + (x_1 + 21) \ge 70 \implies 4x_1 + 41 \ge 70 \implies 4x_1 \ge 29 \implies x_1 \ge 7.25

. Since

x_1

must be an integer, the minimum possible value is

x_1 = 8

. This yields a minimum largest value of

x_7 = 8 + 22 = 30

.

Solving the inequality system to find the smallest valid integer boundaries.

Key Concept

Descriptive Statistics and Optimization

Question 1791Question →

For all real values of $x$ and $y$ , the expression $4x^2 - 9y^2 - 12y - 4$ can be factored into the form $(2x + ay + b)(2x - cy - d)$ , where $a, b, c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 10

Answer

The value of

a + b + c + d

is 10.

Grouping the

y

terms gives

4x^2 - (9y^2 + 12y + 4)

. Factoring the quadratic within the parentheses gives

4x^2 - (3y + 2)^2

. Applying the difference of squares identity

(u^2 - v^2) = (u + v)(u - v)

leads to

(2x + (3y + 2))(2x - (3y + 2)) = (2x + 3y + 2)(2x - 3y - 2)

. Matching this expression to the template

(2x + ay + b)(2x - cy - d)

reveals that

a = 3

,

b = 2

,

c = 3

, and

d = 2

, all of which are positive integers. The sum of these values is

3 + 2 + 3 + 2 = 10

.

Step-by-Step Solution

1

Group the terms containing

y

4x^2 - (9y^2 + 12y + 4)

Grouping the terms allows us to identify a perfect square trinomial pattern.

2

Factor the trinomial inside the parentheses

4x^2 - (3y + 2)^2

The expression

9y^2 + 12y + 4

is a perfect square trinomial of the form

(3y)^2 + 2(3y)(2) + 2^2

.

3

Apply the difference of squares identity

(2x + (3y + 2))(2x - (3y + 2))

The expression is in the form

u^2 - v^2

, where

u = 2x

and

v = 3y + 2

. Using

u^2 - v^2 = (u + v)(u - v)

factors the expression.

4

Simplify the factored binomials by distributing signs

(2x + 3y + 2)(2x - 3y - 2)

Simplifying the expressions removes inner parentheses, allowing comparison with the target template.

5

Compare with the template

(2x + ay + b)(2x - cy - d)

to find the constants

a = 3

,

b = 2

,

c = 3

,

d = 2

Matching the terms directly gives

ay = 3y \Rightarrow a = 3

,

b = 2

,

-cy = -3y \Rightarrow c = 3

, and

-d = -2 \Rightarrow d = 2

.

6

Calculate the sum of the positive integers

10

The question asks for the value of

a + b + c + d

, which is

3 + 2 + 3 + 2 = 10

.

Key Concept

Factoring polynomials using grouping, perfect square trinomials, and the difference of squares identity

Estimated Time:2m 0s

Question 1792Question →

When the expression $2x(x^2 - 3xy + 2y^2) - (x - 2y)^3 - y(x^2 - 5xy + 4y^2)$ is fully simplified by combining like terms, what is the coefficient of the $xy^2$ term?

Show answer & explanation

Answer: -3

Answer

The coefficient of the

xy^2

term is

-3

.

Expanding the terms of the expression yields

2x^3 - 6x^2y + 4xy^2

,

-x^3 + 6x^2y - 12xy^2 + 8y^3

, and

-x^2y + 5xy^2 - 4y^3

. Summing the coefficients of the

xy^2

terms gives

4 - 12 + 5 = -3

. Thus, the coefficient of

xy^2

is

-3

.

Step-by-Step Solution

1

Expand the first term of the expression:

2x(x^2 - 3xy + 2y^2)

2x^3 - 6x^2y + 4xy^2

Distribute the monomial

2x

to each term inside the parentheses:

2x \cdot x^2 = 2x^3

,

2x \cdot (-3xy) = -6x^2y

, and

2x \cdot 2y^2 = 4xy^2

.

2

Expand the cubed binomial

(x - 2y)^3

and distribute the negative sign

-x^3 + 6x^2y - 12xy^2 + 8y^3

First, expand the binomial

(x - 2y)^3 = x^3 - 3(x^2)(2y) + 3(x)(2y)^2 - (2y)^3 = x^3 - 6x^2y + 12xy^2 - 8y^3

. Then, distribute the negative sign to all terms inside the parentheses.

3

Expand the third term:

-y(x^2 - 5xy + 4y^2)

-x^2y + 5xy^2 - 4y^3

Distribute the negative monomial

-y

to each term inside the parentheses:

-y \cdot x^2 = -x^2y

,

-y \cdot (-5xy) = 5xy^2

, and

-y \cdot 4y^2 = -4y^3

.

4

Group and combine all like terms

x^3 - x^2y - 3xy^2 + 4y^3

Combine the coefficients of the corresponding terms:

(2x^3 - x^3) + (-6x^2y + 6x^2y - x^2y) + (4xy^2 - 12xy^2 + 5xy^2) + (8y^3 - 4y^3) = x^3 - x^2y - 3xy^2 + 4y^3

.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:3m 0s

Question 1793Question →

For a certain positive number $y$ , the product of $y$ and the quantity $2y + 5$ is equal to $12$ . What is the value of $y$ ?

Show answer & explanation

Answer: 1.5

Answer

The positive value of

y

is

1.5

.

The correct answer is

1.5

. By translating the word problem, we obtain

y(2y + 5) = 12

. Expanding this gives

2y^2 + 5y = 12

, and subtracting

12

from both sides yields the standard quadratic equation

2y^2 + 5y - 12 = 0

. Factoring this expression gives

(2y - 3)(y + 4) = 0

. Setting the first factor to zero yields

y = 1.5

, which is positive and therefore satisfies the given condition.

Step-by-Step Solution

1

Write the equation representing the relationship.

y(2y + 5) = 12

To translate the verbal description into an algebraic equation.

2

Distribute

y

and set the equation equal to zero.

2y^2 + 5y - 12 = 0

To put the quadratic equation into standard form

ay^2 + by + c = 0

so it can be factored.

3

Factor the quadratic equation.

(2y - 3)(y + 4) = 0

To find the factors that multiply to give the quadratic expression.

4

Solve for

y

and apply the constraint.

y = 1.5

Setting the factors to zero gives

y = 1.5

and

y = -4

. Since the problem states

y

is positive, we select the positive root.

Key Concept

Solving a non-monic quadratic equation by factoring after translating a verbal description into algebra.

Question 1794Question →

A list of $15$ positive integers is sorted in non-decreasing order. The minimum value in the list is $10$ , the maximum value is $50$ , and the median is $30$ . If the unique mode of the list is $20$ , what is the maximum possible sum of the $15$ integers in the list?

488

503

506

508

518

Show answer & explanation

Answer: 508

Answer

508

The maximum sum is achieved by setting the frequency of the unique mode

20

to its maximum limit of

6

, occupying positions

2

through

7

(since the first element is

10

and the median at position

8

is

30

). This allows any other value to appear at most

5

times. To maximize the sum, we assign the maximum value

50

to

5

positions (positions

11

to

15

), and the next highest value

49

to the remaining

2

positions in the upper half (positions

9

and

10

). Summing these values gives

10 + 6(20) + 30 + 2(49) + 5(50) = 508

.

Step-by-Step Solution

1

Identify fixed elements based on the median, minimum, and maximum constraints.

For

15

sorted integers, the median is the 8th term:

x_8 = 30

. The minimum is

x_1 = 10

, and the maximum is

x_{15} = 50

.

Establishing these boundary points helps constrain the values of the other elements in the sorted list.

2

Determine the maximum possible frequency of the unique mode

20

.

Since

x_8 = 30

and the list is sorted,

20

can only occupy positions

2

through

7

(a maximum of

6

occurrences).

To maximize the overall sum, we want to maximize the frequency of the mode so that other large values (like

50

) can also appear as many times as possible.

3

Determine the maximum allowed frequency of other values under the unique mode constraint.

With the frequency of

20

set to

6

, any other number can appear at most

5

times.

Since

20

is the unique mode, no other value in the list can have a frequency equal to or greater than

6

.

4

Assign the largest possible values to the remaining positions in the upper half (

x_9

to

x_{14}

).

Set

x_{11} = x_{12} = x_{13} = x_{14} = x_{15} = 50

(frequency of

5

). The remaining two positions,

x_9

and

x_{10}

, are set to the next largest integer,

49

.

This maximizes the sum of the upper half of the list without violating the maximum frequency limit of

5

for any non-mode value.

5

Sum the elements of the optimal list.

Sum =

10 + 6(20) + 30 + 2(49) + 5(50) = 10 + 120 + 30 + 98 + 250 = 508

.

Adding all fifteen optimal values together gives the absolute maximum possible sum.

Key Concept

Using central tendency constraints (median, mode) and range limits to optimize a dataset sum.

Question 1795Question →

When the polynomial $4x^4 - 37x^2 + 9$ is factored completely into linear factors of the form $ax + b$ , where $a$ and $b$ are integers and $a > 0$ , which of the following expressions represents the sum of these linear factors?

4x

6x - 8

6x

6x^4

6x - 6

Show answer & explanation

Answer: $6x$

Answer

The sum of the linear factors is

6x

.

Factoring the polynomial

4x^4 - 37x^2 + 9

by substituting

u = x^2

yields

(4u - 1)(u - 9)

, which becomes

(4x^2 - 1)(x^2 - 9)

. Applying the difference of squares identity to both terms results in the four linear factors

(2x - 1)

,

(2x + 1)

,

(x - 3)

, and

(x + 3)

. The sum of these factors is

6x

.

Step-by-Step Solution

1

Substitute

u = x^2

to rewrite the quartic polynomial as a quadratic expression.

4u^2 - 37u + 9

This simplifies the polynomial from degree 4 to degree 2, making it easier to factor.

2

Factor the quadratic expression by grouping or finding two numbers that multiply to

36

and add to

-37

.

(4u - 1)(u - 9)

The numbers are

-36

and

-1

. Rewriting and grouping gives

4u(u - 9) - 1(u - 9) = (4u - 1)(u - 9)

.

3

Substitute

x^2

back in place of

u

and factor the resulting difference of squares binomials.

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

Since

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

and

x^2 - 9 = x^2 - 3^2

, both binomials can be factored completely using the difference of squares identity.

4

Sum the four linear factors.

6x

Combining the like terms gives

(2x - 1) + (2x + 1) + (x - 3) + (x + 3) = (2x + 2x + x + x) + (-1 + 1 - 3 + 3) = 6x + 0 = 6x

.

Key Concept

Factoring quartic polynomials using quadratic substitution and the difference of squares identity.

Question 1796Question →

When the expression $2x^2(3x - y) - 3y(x^2 - 2y) - (x^3 - 3xy^2)$ is fully simplified by combining like terms, which of the following represents the resulting expression?

5x^3 - 5x^2y + 3xy^2 + 6y^2

5x^3 - 5x^2y - 3xy^2 - 6y^2

-x^3 + 6x^2 - 5x^2y + 3xy^2 + 6y^2

5x^3 - 5x^2y + 3xy + 6y

5x^3 - x^2y + 3xy^2 + 6y^2

Show answer & explanation

Answer: $5x^3 - 5x^2y + 3xy^2 + 6y^2$

Answer

5x^3 - 5x^2y + 3xy^2 + 6y^2

The correct expression is obtained by systematically distributing the coefficients outside the parentheses and then grouping and combining the coefficients of the like terms:

(6x^3 - x^3) + (-2x^2y - 3x^2y) + 3xy^2 + 6y^2 = 5x^3 - 5x^2y + 3xy^2 + 6y^2

.

Step-by-Step Solution

1

Distribute

2x^2

to each term in the first parenthetical expression:

(3x - y)

6x^3 - 2x^2y

Applying the distributive property of multiplication over subtraction.

2

Distribute

-3y

to each term in the second parenthetical expression:

(x^2 - 2y)

-3x^2y + 6y^2

Applying the distributive property and multiplying negative coefficients (

-3 \times -2 = 6

).

3

Distribute the negative sign to each term in the third parenthetical expression:

(x^3 - 3xy^2)

-x^3 + 3xy^2

Distributing

-1

across the parentheses to remove them.

4

Combine the expanded expressions and group the like terms

(6x^3 - x^3) + (-2x^2y - 3x^2y) + 3xy^2 + 6y^2

Grouping together terms that have the same variables raised to the same powers.

5

Simplify by performing the operations on the coefficients of the like terms

5x^3 - 5x^2y + 3xy^2 + 6y^2

Combining coefficients:

6 - 1 = 5

for the

x^3

terms, and

-2 - 3 = -5

for the

x^2y

terms.

Key Concept

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Alternative Method

To check your work, substitute simple values for

x

and

y

, such as

x = 1

and

y = 1

, into the original expression and the simplified expression. Evaluating the original expression:

2(1)^2(3(1) - 1) - 3(1)(1^2 - 2(1)) - (1^3 - 3(1)(1)^2) = 2(2) - 3(-1) - (1 - 3) = 4 + 3 - (-2) = 9

. Evaluating the correct simplified expression:

5(1)^3 - 5(1)^2(1) + 3(1)(1)^2 + 6(1)^2 = 5 - 5 + 3 + 6 = 9

. Since both evaluations yield 9, this confirms the simplification.

Estimated Time:1m 0s

Question 1797Question →

Given that $i = \sqrt{-1}$ , what is the value of the expression $i^{83}$ ?

i

-1

-i

1

0

Show answer & explanation

Answer: $-i$

Answer

The value of the expression is

-i

.

The expression

i^{83}

can be simplified by dividing the exponent 83 by 4. Because the remainder is 3, the expression is equivalent to

i^3

. Since

i^3 = i^2 \cdot i = (-1) \cdot i = -i

, the correct value is

-i

.

Step-by-Step Solution

1

Divide the exponent 83 by 4 to determine the remainder.

The quotient is 20 with a remainder of 3, which means

83 = 4 \times 20 + 3

.

Powers of the imaginary unit

i

repeat in a cycle of four:

i^1 = i

,

i^2 = -1

,

i^3 = -i

, and

i^4 = 1

.

2

Rewrite the expression using the rules of exponents and substitute the values of

i^4

and

i^3

.

i^{83} = i^{4(20) + 3} = (i^4)^{20} \cdot i^3 = (1)^{20} \cdot (-i) = -i

.

Since

i^4 = 1

, any power of

i

that is a multiple of 4 simplifies to 1, leaving only the remainder power to determine the final value.

Key Concept

Simplifying powers of the imaginary unit

i

Question 1798Question →

A business analyst models the daily cost, $C$ , and daily revenue, $R$ , of a manufacturing process using the following expressions:

C = 2x^2(x - 3y) - y(x^2 - 2y^2)

R = 5x^3 - x(y - 2x)^2 + 3y^3

where $x$ represents the number of units of product X sold, and $y$ represents the number of units of product Y sold. The daily profit, $P$ , is defined as $P = R - C$ . When the profit expression is fully simplified by combining like terms, which of the following expressions represents the daily profit $P$ ?

$-x^3 + 7x^2y - xy^2 + y^3

$-x^3 + 11x^2y - xy^2 + 5y^3

$-x^3 + 11x^2y - xy^2 + y^3

$-x^3 + 3x^2y - xy^2 + y^3

$3x^3 - 4x^2 + 11x^2y - xy^2 + y^3

Show answer & explanation

Answer: $-x^3 + 11x^2y - xy^2 + y^3

Answer

-x^3 + 11x^2y - xy^2 + y^3

The expression

-x^3 + 11x^2y - xy^2 + y^3

is correct. Calculating profit requires subtracting the fully simplified cost from the fully simplified revenue. Simplifying

C

yields

2x^3 - 7x^2y + 2y^3

. Expanding and simplifying

R

yields

x^3 + 4x^2y - xy^2 + 3y^3

. Subtracting these two expressions and correctly distributing the negative sign gives

-x^3 + 11x^2y - xy^2 + y^3

.

Step-by-Step Solution

1

Expand and simplify the cost expression

C = 2x^2(x - 3y) - y(x^2 - 2y^2)

.

C = 2x^3 - 6x^2y - x^2y + 2y^3 = 2x^3 - 7x^2y + 2y^3

Distribute the terms outside the parentheses and combine the like terms

-6x^2y

and

-x^2y

.

2

Expand and simplify the revenue expression

R = 5x^3 - x(y - 2x)^2 + 3y^3

.

R = 5x^3 - x(y^2 - 4xy + 4x^2) + 3y^3 = 5x^3 - xy^2 + 4x^2y - 4x^3 + 3y^3 = x^3 + 4x^2y - xy^2 + 3y^3

Square the binomial

(y - 2x)^2 = y^2 - 4xy + 4x^2

, distribute the

-x

, and combine the like terms

5x^3

and

-4x^3

.

3

Subtract the cost expression from the revenue expression to find

P = R - C

.

P = (x^3 + 4x^2y - xy^2 + 3y^3) - (2x^3 - 7x^2y + 2y^3) = x^3 + 4x^2y - xy^2 + 3y^3 - 2x^3 + 7x^2y - 2y^3 = -x^3 + 11x^2y - xy^2 + y^3

Distribute the subtraction negative sign to all terms in the cost expression and group like terms together to obtain the final simplified expression.

Key Concept

Simplifying multi-variable algebraic expressions by distributing coefficients and combining like terms.

Question 1799Question →

Match each algebraic expression on the left with its fully simplified equivalent expression on the right. All variables represent real numbers.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The first expression

3a(a^2 - 2ab) - (a^3 - 5a^2b)

matches

2a^3 - a^2b

. The second expression

a^2(2a - b) - 2a(a^2 - ab)

matches

a^2b

. The third expression

-(a^3 + 3a^2b) + 3a^2(a + b)

matches

2a^3

.

Each expression on the left is simplified by distributing the coefficients and combining the like terms. The first expression simplifies to

2a^3 - a^2b

. The second expression simplifies to

a^2b

. The third expression simplifies to

2a^3

. These match the corresponding simplified expressions on the right.

Step-by-Step Solution

1

Simplify the first expression

3a(a^2 - 2ab) - (a^3 - 5a^2b)

2a^3 - a^2b

Multiply

3a

by both terms in the first parentheses to get

3a^3 - 6a^2b

. Distribute the negative sign to both terms in the second parentheses to get

-a^3 + 5a^2b

. Combine the

a^3

terms to get

2a^3

and the

a^2b

terms to get

-a^2b

.

2

Simplify the second expression

a^2(2a - b) - 2a(a^2 - ab)

a^2b

Multiply

a^2

by both terms in the first parentheses to get

2a^3 - a^2b

. Multiply

-2a

by both terms in the second parentheses to get

-2a^3 + 2a^2b

. Combine the

a^3

terms to get

0

and the

a^2b

terms to get

a^2b

.

3

Simplify the third expression

-(a^3 + 3a^2b) + 3a^2(a + b)

2a^3

Distribute the negative sign to get

-a^3 - 3a^2b

. Multiply

3a^2

by both terms in the second parentheses to get

3a^3 + 3a^2b

. Combine the

a^3

terms to get

2a^3

and the

a^2b

terms to get

0

.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

Question 1800Question →

Determine the value of $x$ for which the given rational expression is undefined.

Fill in the blanks below

The rational expression

\frac{x + 7}{2x - 10}

is undefined when

x =

.

Show answer & explanation

Answer

5

The rational expression is undefined when the denominator,

2x - 10

, equals

0

. Solving

2x - 10 = 0

gives

2x = 10

, which simplifies to

x = 5

.

Step-by-Step Solution

1

Identify the denominator of the rational expression and set it equal to zero.

2x - 10 = 0

A rational expression is undefined when its denominator is equal to zero, as division by zero is not defined in the real number system.

2

Solve the linear equation for

x

by adding 10 to both sides and then dividing by 2.

2x = 10 \Rightarrow x = 5

Isolating

x

yields the value that makes the denominator zero.

Key Concept

Rational expressions are undefined when the denominator is equal to zero.

Estimated Time:45s

Answer

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