All practice questions

Question 81Question →

If the expression $6x^2 - 11x - 10$ is factored completely into the form $(ax + b)(cx - d)$ , where $a, b, c,$ and $d$ are positive integers, what is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 12

Answer

The value of the sum of the coefficients and constants

a + b + c + d

is

12

.

The factored form of

6x^2 - 11x - 10

is

(3x + 2)(2x - 5)

. Matching this with

(ax + b)(cx - d)

where

a, b, c,

and

d

are positive integers results in

a = 3

,

b = 2

,

c = 2

, and

d = 5

. The sum of these values is

3 + 2 + 2 + 5 = 12

.

Step-by-Step Solution

1

Factor the quadratic trinomial

6x^2 - 11x - 10

using the grouping method.

(3x + 2)(2x - 5)

Factoring splits the quadratic expression into its constituent linear binomial factors.

2

Equate the factored expression to the given form

(ax + b)(cx - d)

to find the values of

a, b, c,

and

d

.

a = 3

,

b = 2

,

c = 2

, and

d = 5

Since the variables represent positive integers, we match the positive constant term to

b

and the negative constant term to

-d

.

3

Sum the values of

a, b, c,

and

d

.

12

Calculating the final sum answers the target mathematical question.

Key Concept

Factoring quadratic polynomials with a leading coefficient greater than 1 using the grouping method.

Question 82Question →

The cubic polynomial $6x^3 - 19x^2 + 11x + 6$ can be factored completely over the integers in the form $(x - a)(bx - c)(dx + e)$ , where $a$ , $b$ , $c$ , $d$ , and $e$ are positive integers. What is the value of $a + b + c + d + e$ ?

Show answer & explanation

Answer: 11

Answer

The sum of the coefficients and constants from the factored form is 11.

By applying the Rational Root Theorem, we find the root

x = 2

, which gives the factor

(x - 2)

. Dividing the original cubic expression by

(x - 2)

yields

6x^2 - 7x - 3

. Factoring this quadratic expression by grouping yields

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

. Writing the completely factored form as

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

and comparing it to

(x - a)(bx - c)(dx + e)

where

a, b, c, d, e

are positive integers results in

a = 2

,

b = 2

,

c = 3

,

d = 3

, and

e = 1

. The sum

a + b + c + d + e

is equal to 11.

Step-by-Step Solution

1

Identify a rational root of

6x^3 - 19x^2 + 11x + 6

x = 2

is a root, meaning

(x - 2)

is a factor.

Applying the Rational Root Theorem and testing potential integer root values.

2

Divide

6x^3 - 19x^2 + 11x + 6

by

(x - 2)

The quotient is the quadratic polynomial

6x^2 - 7x - 3

.

To reduce the cubic polynomial to a quadratic expression that can be factored using standard trinomial methods.

3

Factor the quadratic trinomial

6x^2 - 7x - 3

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

Finding two numbers that multiply to

-18

and add to

-7

(which are

-9

and

2

) and factoring by grouping.

4

Match the factored form

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

to

(x - a)(bx - c)(dx + e)

a = 2, b = 2, c = 3, d = 3, e = 1

Since

a, b, c, d,

and

e

must be positive integers, the constant terms and signs uniquely determine the mapping of each factor.

5

Calculate the sum of

a, b, c, d,

and

e

2 + 2 + 3 + 3 + 1 = 11

To find the final numeric value requested by the question.

Key Concept

Factoring cubic polynomials by finding rational roots and factoring quadratic trinomials by grouping.

Estimated Time:2m 30s

Question 83Question →

If $f(x) = 3x - 2$ and $g(x) = x^2 + 1$ , what is the value of $g(f(2))$ ?

Show answer & explanation

Answer: 17

Answer

The value of

g(f(2))

is 17.

To evaluate the composite function

g(f(2))

, we first find the value of the inner function

f(2) = 3(2) - 2 = 4

. Then, we substitute this output as the input for the outer function to get

g(4) = 4^2 + 1 = 17

.

Step-by-Step Solution

1

Substitute

x = 2

into the expression for

f(x)

to evaluate the inner function.

f(2) = 4

In the composition

g(f(2))

, the inner function

f(x)

must be evaluated first at the input value of 2.

2

Substitute the result from Step 1, which is 4, into the expression for

g(x)

.

g(4) = 17

The output of the inner function becomes the input for the outer function, so

g(f(2)) = g(4) = 4^2 + 1 = 17

.

Key Concept

Function Composition

Estimated Time:45s

Question 84Question →

For what value of $a$ does the equation $\frac{3}{4}(x - 2) - \frac{1}{3}(2x + a) = \frac{1}{12}x - 5$ have infinitely many solutions for $x$ ?

Show answer & explanation

Answer: 10.5

Answer

10.5

Expanding the left side of the equation yields

\frac{3}{4}x - \frac{3}{2} - \frac{2}{3}x - \frac{a}{3}

. Combining the coefficients of

x

gives

(\frac{3}{4} - \frac{2}{3})x = \frac{1}{12}x

. The equation becomes

\frac{1}{12}x - (\frac{3}{2} + \frac{a}{3}) = \frac{1}{12}x - 5

. For a linear equation to have infinitely many solutions, the variable coefficients must be equal and the constant terms must also be equal. Therefore, we equate the constants:

-\frac{3}{2} - \frac{a}{3} = -5

. Multiplying all terms by

-6

to clear the denominators yields

9 + 2a = 30

, which simplifies to

2a = 21

, or

a = 10.5

.

Step-by-Step Solution

1

Expand and simplify the left side of the equation by distributing the fraction coefficients.

\frac{1}{12}x - \frac{3}{2} - \frac{a}{3}

Distributing

\frac{3}{4}

and

-\frac{1}{3}

across their respective parentheses and combining the

x

terms allows us to compare the coefficients on both sides.

2

Equate the constant terms from both sides of the equation.

-\frac{3}{2} - \frac{a}{3} = -5

A linear equation of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

. Since both

A

and

C

are

\frac{1}{12}

, we set the constant terms equal.

3

Isolate the variable

a

and solve.

a = 10.5

Adding

\frac{3}{2}

to both sides gives

-\frac{a}{3} = -3.5

. Multiplying both sides by

-3

yields the final value.

Key Concept

Solving linear equations with infinitely many solutions by equating coefficients and constant terms.

Estimated Time:2m 0s

Question 85Question →

A chemist is preparing $120\text{ milliliters}$ of a chemical mixture with an overall acid concentration of $27.5\%$ by volume. To do this, she mixes Solution $X$ ( $10\%$ acid by volume), Solution $Y$ ( $25\%$ acid by volume), and Solution $Z$ ( $40\%$ acid by volume). The chemist decides that the volume of Solution $Y$ must be exactly $20\text{ milliliters}$ less than twice the volume of Solution $X$ used in the mixture. What is the positive difference, in milliliters, between the volume of Solution $Z$ and the volume of Solution $X$ in the final mixture?

Show answer & explanation

Answer: 20

Answer

The positive difference between the volume of Solution Z and Solution X is 20 milliliters.

The correct answer is 20, because solving the system of equations yields a volume of 30 milliliters for Solution X and 50 milliliters for Solution Z. The positive difference between these two volumes is 20 milliliters.

Step-by-Step Solution

1

Define variables for the volume of each solution used in the mixture.

Let

x

be the volume of Solution

X

,

y

be the volume of Solution

Y

, and

z

be the volume of Solution

Z

(all in milliliters).

Defining variables allows for translating the word problem's conditions into algebraic equations.

2

Translate the given information into a system of three linear equations.

Equation 1 (Total Volume):

x + y + z = 120

. Equation 2 (Total Acid):

0.10x + 0.25y + 0.40z = 33

(since

27.5\%

of

120\text{ ml}

is

33\text{ ml}

). Equation 3 (Volume Relation):

y = 2x - 20

.

These equations model the constraints and quantities described in the problem.

3

Express

z

in terms of

x

by substituting the expression for

y

into the total volume equation.

x + (2x - 20) + z = 120 \implies 3x - 20 + z = 120 \implies z = 140 - 3x

.

This substitution reduces the system to two variables (

x

and

z

), making it easier to solve.

4

Substitute the expressions for

y

and

z

in terms of

x

into the acid equation and solve for

x

.

0.10x + 0.25(2x - 20) + 0.40(140 - 3x) = 33 \implies 0.10x + 0.50x - 5 + 56 - 1.20x = 33 \implies -0.60x + 51 = 33 \implies -0.60x = -18 \implies x = 30

.

This isolates the single variable

x

so that its value can be calculated.

5

Determine the volume of Solution

Z

and calculate the final positive difference.

z = 140 - 3(30) = 50

. The positive difference between the volume of Solution

Z

and Solution

X

is

|50 - 30| = 20

.

This directly answers the question's requirement for the difference between the two solution volumes.

Key Concept

Translating word problems into a system of three linear equations and solving them using substitution.

Question 86Question →

An art gallery owner wants to display 5 paintings in a row on a wall. The owner selects these 5 paintings from a collection of 4 different landscape paintings and 4 different portrait paintings. The display must meet the following guidelines:

1. No two landscape paintings can be placed next to each other.
2. At least 2 landscape paintings must be displayed.

How many different arrangements of 5 paintings are possible?

Show answer & explanation

Answer: 2016

Answer

2016

The correct answer is 2016. By breaking down the problem into two mutually exclusive cases based on the number of landscape paintings (either 3 landscapes and 2 portraits, or 2 landscapes and 3 portraits), we can find the valid arrangements for each. For 3 landscapes, the only valid layout is

L-P-L-P-L

, yielding

P(4,3) \times P(4,2) = 288

arrangements. For 2 landscapes, there are

\binom{4}{2} = 6

layout configurations, each yielding

P(4,2) \times P(4,3) = 288

arrangements, for a total of 1728. Adding these two cases gives

288 + 1728 = 2016

.

Step-by-Step Solution

1

Determine the possible number of landscape paintings (

k

) that can be displayed.

k = 2

or

k = 3

Since at least 2 landscapes must be displayed,

k \ge 2

. Since no two landscapes can be adjacent in a 5-painting row, we cannot have 4 landscapes (as that would require at least 3 portraits to separate them, making the total count at least 7 paintings). Thus,

k

can only be 2 or 3.

2

Calculate the arrangements for the case with 3 landscape paintings and 2 portrait paintings.

288 arrangements

For 3 landscapes (

L

) and 2 portraits (

P

) to have no adjacent landscapes, the only possible pattern of positions is

L-P-L-P-L

. The number of ways to choose and arrange 3 landscapes from 4 is

P(4, 3) = 4 \times 3 \times 2 = 24

. The number of ways to choose and arrange 2 portraits from 4 is

P(4, 2) = 4 \times 3 = 12

. The total arrangements for this case is

24 \times 12 = 288

.

3

Calculate the arrangements for the case with 2 landscape paintings and 3 portrait paintings.

1728 arrangements

For 2 landscapes (

L

) and 3 portraits (

P

) to have no adjacent landscapes, we place the 3 portraits first:

\_ P \_ P \_ P \_

. We choose 2 of the 4 available spaces for the landscapes in

\binom{4}{2} = 6

ways. For each pattern, the number of ways to choose and arrange 2 landscapes from 4 is

P(4, 2) = 12

, and the number of ways to choose and arrange 3 portraits from 4 is

P(4, 3) = 24

. The total arrangements for this case is

6 \times 12 \times 24 = 1728

.

4

Sum the arrangements from both cases.

2016 arrangements

Since the two cases are mutually exclusive, we add their individual counts:

288 + 1728 = 2016

.

Key Concept

Permutations and Combinations with Constraints

Question 87Question →

A manufacturer of custom planners determines that the setup cost for a production run is $100$ dollars, and each planner costs $6$ dollars to produce. The planners sell for $10$ dollars each, except for the first $10$ planners sold, which are discounted by $2$ dollars each. If the manufacturer wants to achieve a net profit of exactly $300$ dollars for a single production run, how many planners must they produce and sell?

Show answer & explanation

Answer: 105

Answer

The manufacturer must produce and sell

105

planners to achieve a net profit of

300

dollars.

The correct answer is found by setting up the profit equation:

\text{Profit} = \text{Total Revenue} - \text{Total Cost}

. The cost is

100 + 6x

and the revenue is

10(8) + 10(x-10) = 10x - 20

. Equating their difference to

300

yields

(10x - 20) - (100 + 6x) = 300

, which simplifies to

4x - 120 = 300

. Solving for

x

results in

105

.

Step-by-Step Solution

1

Define the variable for the number of planners.

Let

x

be the number of planners produced and sold, where

x \geq 10

.

Establishing the variable is necessary to set up algebraic expressions for cost and revenue.

2

Write the total cost expression.

\text{Total Cost} = 100 + 6x

The cost combines the fixed setup fee of

100

dollars and the variable cost of

6

dollars per planner.

3

Write the total revenue expression.

\text{Total Revenue} = 10(8) + 10(x - 10) = 10x - 20

The first

10

planners sell for

8

dollars each, and the remaining

x - 10

planners sell for the regular price of

10

dollars each.

4

Set up the profit equation and solve for

x

.

(10x - 20) - (100 + 6x) = 300 \implies 4x - 120 = 300 \implies 4x = 420 \implies x = 105

Profit is the difference between total revenue and total cost, which must equal the target profit of

300

dollars.

Key Concept

Translating real-world pricing and cost constraints into a single-variable linear equation.

Question 88Question →

For $x \neq 0$ , the expression $\frac{(x^2)^5}{x^k}$ simplifies to $x^6$ . What is the value of the exponent $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the exponent

k

is 4.

Applying the power of a power rule to the numerator yields

(x^2)^5 = x^{10}

. Next, applying the quotient rule to divide

x^{10}

by

x^k

yields

x^{10-k}

. Setting this equal to the simplified term

x^6

leads to the exponent equation

10 - k = 6

. Solving this equation gives the final result

k = 4

.

Step-by-Step Solution

1

Simplify the numerator expression

(x^2)^5

x^{10}

Multiply the exponents when raising a power to another power:

(x^a)^b = x^{ab}

.

2

Simplify the division of the two exponential expressions

\frac{x^{10}}{x^k}

x^{10-k}

Subtract the exponent of the denominator from the exponent of the numerator:

\frac{x^a}{x^b} = x^{a-b}

.

3

Solve the linear equation for

k

using the target exponent 6

k = 4

Equating the exponent

10-k

to

6

gives

10-k=6

. Subtracting 10 from both sides gives

-k = -4

, so

k = 4

.

Key Concept

Applying properties of exponents in algebraic expressions, specifically the power of a power rule and the quotient rule.

Question 89Question →

For what value of the constant $a$ does the linear equation $\frac{a(x - 2)}{3} - \frac{3x - 1}{2} = -\frac{5x + 7}{6}$ have no real solution for $x$ ?

Show answer & explanation

Answer: 2

Answer

The constant

a

must be equal to

2

for the equation to have no solution.

A linear equation in the form

Ax + B = Cx + D

has no solution if the coefficients of the variable are equal (

A = C

) but the constant terms are different (

B \neq D

). Multiplying the given equation by the least common denominator, 6, clears the fractions and yields

2a(x - 2) - 3(3x - 1) = -(5x + 7)

. Expanding both sides results in

2ax - 4a - 9x + 3 = -5x - 7

, which simplifies to

(2a - 9)x + (3 - 4a) = -5x - 7

. Equating the coefficients of

x

gives

2a - 9 = -5

, which solves to

a = 2

. Substituting

a = 2

back into the constants yields a left-side constant of

-5

and a right-side constant of

-7

. Since

-5 \neq -7

, the variable terms cancel out while leaving an inequality, meaning the equation has no solution when

a = 2

.

Step-by-Step Solution

1

Clear the denominators by multiplying the entire equation by the least common denominator, which is 6.

2a(x - 2) - 3(3x - 1) = -(5x + 7)

Multiplying by the LCD simplifies the rational expressions into polynomial terms.

2

Distribute and expand the terms on both sides of the equation.

2ax - 4a - 9x + 3 = -5x - 7

Expanding the terms allows us to group variable terms and constant terms together.

3

Group the

x

terms and constant terms on the left side.

(2a - 9)x + (3 - 4a) = -5x - 7

Structuring the equation in the standard form

Ax + B = Cx + D

makes it easier to compare coefficients.

4

Set the coefficients of

x

on both sides equal to each other.

2a - 9 = -5

For a linear equation to have no solution, the variable terms must cancel out, meaning their coefficients must be identical.

5

Solve for the parameter

a

and verify the constant terms are unequal.

2a = 4 \implies a = 2

. Constant check:

3 - 4(2) = -5

, and

-5 \neq -7

.

If the constant terms were equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Identifying the parameter value that results in a linear equation having no solution by equating variable coefficients and ensuring constant terms are unequal.

Question 90Question →

If the algebraic expression $\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}$ simplifies to $x^4 y^4$ for all non-zero real numbers $x$ and $y$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a + b

is

4

.

Applying exponent rules to the expression

\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}

yields

\frac{x^{-3a}y^{-6} \cdot x^4 y^{2b}}{x^6 y^2} = \frac{x^{-3a+4} y^{2b-6}}{x^6 y^2} = x^{-3a-2} y^{2b-8}

. Equating these exponents to the target expression

x^4 y^4

gives the system

-3a - 2 = 4

and

2b - 8 = 4

. Solving these equations gives

a = -2

and

b = 6

, resulting in a sum of

a + b = 4

.

Step-by-Step Solution

1

Simplify the terms in the numerator.

(x^a y^2)^{-3} = x^{-3a} y^{-6}

and

(x^2 y^b)^2 = x^4 y^{2b}

Apply the power of a product rule:

(u^m v^n)^p = u^{mp} v^{np}

.

2

Multiply the simplified terms in the numerator.

x^{-3a+4} y^{2b-6}

Apply the product rule of exponents by adding exponents of like bases:

u^m \cdot u^n = u^{m+n}

.

3

Simplify the denominator.

(x^{-3} y^{-1})^{-2} = x^6 y^2

Apply the power of a product rule.

4

Divide the numerator by the denominator.

x^{-3a-2} y^{2b-8}

Apply the quotient rule of exponents by subtracting denominator exponents from numerator exponents:

\frac{u^m}{u^n} = u^{m-n}

.

5

Set up equations by equating the simplified exponents to the exponents in the target expression

x^4 y^4

.

-3a - 2 = 4

and

2b - 8 = 4

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of

x

and

y

must be equal.

6

Solve the linear equations for the integer constants

a

and

b

.

a = -2

and

b = 6

Isolate the variables:

-3a = 6 \implies a = -2

, and

2b = 12 \implies b = 6

.

7

Find the sum of

a

and

b

.

4

Add the values of the constants:

-2 + 6 = 4

.

Key Concept

Properties of exponents (product, quotient, and power rules) in multi-step algebraic simplification

Question 91Question →

For all real numbers $m$ and $n$ , the expression $4m(2m - 3n)^2 - n(5m^2 - 36mn)$ can be written in the form $am^3 + bm^2n + cmn^2$ , where $a$ , $b$ , and $c$ are real constants. What is the value of the coefficient $b$ ?

Show answer & explanation

Answer: -53

Answer

The value of the coefficient

b

is

-53

.

Expanding the entire expression yields

16m^3 - 53m^2n + 72mn^2

. Comparing this to the template form

am^3 + bm^2n + cmn^2

shows that

b = -53

.

Step-by-Step Solution

1

Expand the squared binomial

(2m - 3n)^2

4m^2 - 12mn + 9n^2

Apply the identity

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

to expand the expression inside the parentheses.

2

Distribute

4m

through the trinomial

16m^3 - 48m^2n + 36mn^2

Multiply each term of

4m^2 - 12mn + 9n^2

by

4m

using properties of exponents.

3

Distribute

-n

across

(5m^2 - 36mn)

-5m^2n + 36mn^2

Multiply

-n

by both terms inside the parentheses, paying attention to sign rules.

4

Group and combine the like terms

16m^3 + (-48m^2n - 5m^2n) + (36mn^2 + 36mn^2)

Identify terms with the same variables and exponents to combine them.

5

Combine the coefficients of the like terms

16m^3 - 53m^2n + 72mn^2

Perform arithmetic on the coefficients:

-48 - 5 = -53

for the

m^2n

terms and

36 + 36 = 72

for the

mn^2

terms.

6

Identify the coefficient

b

-53

Compare the simplified polynomial to the form

am^3 + bm^2n + cmn^2

to find the value of

b

.

Key Concept

Simplifying algebraic expressions by expanding binomials, distributing variables and signs, and combining like terms.

Alternative Method

Instead of expanding the entire expression, focus only on terms that produce

m^2n

: from the first part,

4m \times (-12mn) = -48m^2n

, and from the second part,

-n \times 5m^2 = -5m^2n

. Adding these gives

-53m^2n

.

Estimated Time:1m 30s

Question 92Question →

For a real number $x$ , the equation $2(x + 5) = 16$ is true. What is the value of the expression $3x - 1$ ?

Show answer & explanation

Answer: 8

Answer

The value of the expression

3x - 1

is 8.

Solving the equation

2(x + 5) = 16

gives

x = 3

. Evaluating the expression

3x - 1

for

x = 3

yields

3(3) - 1 = 8

.

Step-by-Step Solution

1

Distribute the 2 on the left side of the equation.

2x + 10 = 16

To eliminate the parentheses using the distributive property.

2

Subtract 10 from both sides of the equation.

2x = 6

To isolate the variable term on one side of the equation.

3

Divide both sides of the equation by 2.

x = 3

To find the value of

x

.

4

Substitute the value of

x

into the expression

3x - 1

.

3(3) - 1 = 8

To evaluate the final expression as requested by the question.

Key Concept

Solving multi-step linear equations and evaluating algebraic expressions.

Estimated Time:45s

Question 93Question →

What is the positive real solution to the equation $\frac{3}{x-2} - \frac{2}{x} = 1$ ?

Show answer & explanation

Answer: 4

Answer

The positive real solution to the equation is 4.

By multiplying the entire equation by the least common denominator,

x(x-2)

, the rational equation is cleared of fractions, resulting in

3x - 2(x-2) = x(x-2)

. Simplifying this yields the quadratic equation

x^2 - 3x - 4 = 0

. Factoring the quadratic equation gives

(x - 4)(x + 1) = 0

, which has roots of

x = 4

and

x = -1

. The positive solution is 4.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator,

x(x-2)

, to eliminate the fractions.

3x - 2(x-2) = x(x-2)

Eliminating the denominators simplifies the rational equation into a polynomial equation.

2

Distribute and combine like terms to simplify both sides of the equation.

x + 4 = x^2 - 2x

Simplification is necessary before rearranging the equation into a solvable form.

3

Rearrange the terms into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 - 3x - 4 = 0

Setting the quadratic expression equal to zero allows for factoring and solving.

4

Factor the quadratic equation.

(x - 4)(x + 1) = 0

Finding the factors reveals the potential values for

x

.

5

Solve for the roots and select the positive real solution.

x = 4

The equation yields two solutions,

x = 4

and

x = -1

, and the question explicitly requests the positive solution.

Key Concept

Solving rational equations by converting them to quadratic equations using the least common denominator.

Question 94Question →

The cubic polynomial $2x^3 + 5x^2 - 8x - 20$ can be factored completely over the integers into the form $(x - a)(x + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 11

Answer

The value of

a + b + c + d

is 11.

The polynomial

2x^3 + 5x^2 - 8x - 20

can be factored completely by first grouping the terms as

x^2(2x + 5) - 4(2x + 5) = (x^2 - 4)(2x + 5)

. Factoring the difference of squares

x^2 - 4

yields

(x - 2)(x + 2)(2x + 5)

. Matching this to the given form

(x - a)(x + b)(cx + d)

gives the positive integers

a = 2

,

b = 2

,

c = 2

, and

d = 5

. The sum of these values is

2 + 2 + 2 + 5 = 11

.

Step-by-Step Solution

1

Group the terms of the polynomial

2x^3 + 5x^2 - 8x - 20

.

(2x^3 + 5x^2) - (8x + 20)

Grouping the terms allows us to look for common factors within each pair of terms.

2

Factor out the greatest common factor (GCF) from each group.

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

The GCF of the first group

2x^3 + 5x^2

is

x^2

, and the GCF of the second group

8x + 20

is

4

.

3

Factor out the common binomial factor

(2x + 5)

.

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

Both terms share the binomial factor

(2x + 5)

.

4

Factor the difference of squares

x^2 - 4

.

(x - 2)(x + 2)(2x + 5)

The term

x^2 - 4

is a difference of squares, which factors as

(x - 2)(x + 2)

.

5

Compare the factored expression with the template

(x - a)(x + b)(cx + d)

to determine the values of

a

,

b

,

c

, and

d

.

a = 2

,

b = 2

,

c = 2

, and

d = 5

Comparing the terms yields

x - a = x - 2 \implies a = 2

,

x + b = x + 2 \implies b = 2

, and

cx + d = 2x + 5 \implies c = 2, d = 5

. All values are positive integers as required.

6

Calculate the sum

a + b + c + d

.

11

Substituting the values of the variables into the expression gives

2 + 2 + 2 + 5 = 11

.

Key Concept

Factoring a cubic polynomial by grouping and then factoring the resulting difference of squares.

Alternative Method

Instead of factoring by grouping, we can use the Rational Root Theorem to find rational roots of the polynomial. The possible rational roots of

2x^3 + 5x^2 - 8x - 20 = 0

are of the form

\pm \frac{p}{q}

, where

p

is a factor of

20

and

q

is a factor of

2

. Testing values shows that

x = 2

and

x = -2

are roots, which corresponds to the linear factors

(x - 2)

and

(x + 2)

. Dividing the original cubic by their product,

(x^2 - 4)

, yields the remaining linear factor

(2x + 5)

.

Estimated Time:1m 30s

Question 95Question →

Let the function $f$ be defined by $f(x) = \frac{x - 1}{x + 1}$ for all real numbers $x \neq -1$ . Let $f^1(x) = f(x)$ , and let $f^n(x) = f(f^{n-1}(x))$ for all integers $n \geq 2$ . What is the value of $30 \cdot f^{2026}(3)$ ?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.

Evaluating successive iterations of

f(3)

reveals a repeating sequence:

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

,

f^4(3) = 3

, and then

f^5(3) = 1/2

. This indicates a cycle of period 4. Dividing the power 2026 by the period 4 gives a remainder of 2, meaning

f^{2026}(3)

is equal to

f^2(3) = -1/3

. Multiplying this by 30 yields the final answer of -10.

Step-by-Step Solution

1

Calculate the first few compositions of the function evaluated at the given input

x = 3

.

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

, and

f^4(3) = 3

.

To look for a repeating pattern or periodic behavior in the iterated function composition.

2

Identify the period of the repeating cycle.

The cycle has a length of 4, repeating the values

[1/2, -1/3, -2, 3]

.

Since

f^4(3) = 3

, evaluating further iterations will yield the same sequence of values.

3

Use modular arithmetic to find the value of the 2026th composition.

2026 \equiv 2 \pmod 4

, meaning

f^{2026}(3) = f^2(3) = -1/3

.

Since the cycle repeats every 4 iterations, dividing 2026 by 4 yields a remainder of 2, indicating the second value in the sequence.

4

Multiply the computed composition value by the given coefficient.

30 \cdot (-1/3) = -10

.

To find the final value of the expression

30 \cdot f^{2026}(3)

.

Key Concept

Evaluating repeated function compositions using periodicity and modular arithmetic.

Question 96Question →

A gardener starts with $15$ flowers already planted in a garden. She plans to plant additional flowers at a constant rate of $8$ flowers per hour. How many hours will it take the gardener to have a total of $79$ flowers planted?

Show answer & explanation

Answer: 8

Answer

It will take the gardener

8

hours to have a total of

79

flowers planted.

The total number of flowers planted can be modeled by the linear equation

15 + 8h = 79

, where

h

is the number of hours. Subtracting

15

from both sides of the equation gives

8h = 64

. Dividing both sides by

8

reveals that

h = 8

hours.

Step-by-Step Solution

1

Set up the linear equation representing the total flowers planted over time.

15 + 8h = 79

, where

h

is the number of hours.

The gardener begins with

15

flowers and adds

8

flowers for each hour

h

, with the final goal of

79

total flowers.

2

Isolate the variable term by subtracting the initial number of flowers from the total.

8h = 64

Subtracting

15

from both sides of the equation isolates the term containing the variable

h

.

3

Divide by the rate to solve for the number of hours.

h = 8

Dividing the remaining flowers to be planted (

64

) by the planting rate (

8

flowers per hour) yields the total number of hours required.

Key Concept

Translating a real-world scenario into a linear equation and solving for the unknown variable.

Question 97Question →

For all non-zero real numbers $x$ , the expression $x^4 \cdot (x^3)^k$ is equivalent to $x^{10}$ . What is the value of the integer $k$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

First, use the power of a power property to write

(x^3)^k

as

x^{3k}

. The expression then becomes

x^4 \cdot x^{3k}

. Next, use the product of powers property to combine the terms into

x^{4+3k}

. Since the expression is equivalent to

x^{10}

, set the exponents equal:

4 + 3k = 10

. Solving this equation gives

3k = 6

, which simplifies to

k = 2

.

Step-by-Step Solution

1

Apply the power of a power rule

(x^a)^b = x^{ab}

to simplify

(x^3)^k

.

x^{3k}

To raise a power to another power, multiply the exponents.

2

Apply the product of powers rule

x^a \cdot x^b = x^{a+b}

to combine the terms

x^4 \cdot x^{3k}

.

x^{4+3k}

When multiplying exponential terms with the same base, add their exponents.

3

Set the combined exponent

4+3k

equal to the target exponent

10

and solve for

k

.

k = 2

Since the bases are equal and non-zero, their exponents must be equal.

Key Concept

Properties of exponents in algebraic expressions (power of a power rule and product of powers rule)

Estimated Time:45s

Question 98Question →

For all non-zero real numbers $x$ and $y$ , the expression

\frac{(x^2 y^{-3})^{-2} (x^{-1} y^4)^3}{(x^3 y^{-2})^d}

can be written in the form $x^p y^q$ , where $p$ and $q$ are integers. If $q = 2p$ , what is the value of $d$ ?

Show answer & explanation

Answer: -4

Answer

-4

Applying the rules of exponents yields the simplified expression

x^{-7-3d} y^{18+2d}

. Setting the exponent of

y

equal to twice the exponent of

x

gives the equation

18+2d = 2(-7-3d)

, which solves to

d = -4

.

Step-by-Step Solution

1

Apply the power of a power rule to the terms in the numerator.

(x^2 y^{-3})^{-2} = x^{-4} y^6

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To raise a power to another power, multiply the exponents:

(u^m)^n = u^{mn}

.

2

Multiply the simplified terms in the numerator together.

x^{-4} y^6 \cdot x^{-3} y^{12} = x^{-7} y^{18}

To multiply powers with the same base, add the exponents:

u^m \cdot u^n = u^{m+n}

.

3

Apply the power of a power rule to the denominator.

(x^3 y^{-2})^d = x^{3d} y^{-2d}

Distribute the exponent

d

to both variables inside the parentheses by multiplying the exponents.

4

Divide the numerator by the denominator.

\frac{x^{-7} y^{18}}{x^{3d} y^{-2d}} = x^{-7-3d} y^{18-(-2d)} = x^{-7-3d} y^{18+2d}

To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{u^m}{u^n} = u^{m-n}

.

5

Set up the linear equation for

d

using

q = 2p

and solve.

18+2d = 2(-7-3d) \implies 18+2d = -14-6d \implies 8d = -32 \implies d = -4

The problem states the relationship between the final exponents is

q = 2p

, where

p = -7-3d

and

q = 18+2d

.

Key Concept

Properties of exponents (product, quotient, and power rules) combined with solving a linear equation.

Question 99Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{a + bi}{1 - 2i}$ , where $a$ and $b$ are real numbers. If $z = 4 + 3i$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is 5.

To find the value of

a + b

, we start with the equation

\frac{a + bi}{1 - 2i} = 4 + 3i

. Multiplying both sides by the denominator gives

a + bi = (4 + 3i)(1 - 2i)

. Expanding the right side using the distributive property, we get

a + bi = 4(1) + 4(-2i) + 3i(1) + 3i(-2i) = 4 - 8i + 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 - 5i - 6(-1) = 4 - 5i + 6 = 10 - 5i

. By comparing the real and imaginary parts of both sides, we find that

a = 10

and

b = -5

. The sum of these two values is

a + b = 10 + (-5) = 5

.

Step-by-Step Solution

1

Isolate the numerator by multiplying both sides by the denominator.

a + bi = (4 + 3i)(1 - 2i)

To solve for the variables

a

and

b

in the numerator, we clear the fraction by multiplying by the denominator.

2

Expand the product of the two complex numbers.

a + bi = 4 - 8i + 3i - 6i^2

Distribute each term of the first binomial to each term of the second binomial.

3

Simplify the expression using the definition of

i^2

.

a + bi = 10 - 5i

Since

i^2 = -1

, the term

-6i^2

becomes

+6

. Combine the real parts (

4 + 6 = 10

) and imaginary parts (

-8i + 3i = -5i

).

4

Equate the components and calculate

a + b

.

a = 10

,

b = -5

, and

a + b = 5

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Therefore,

a = 10

and

b = -5

. Summing these yields

10 + (-5) = 5

.

Key Concept

Equality and multiplication of complex numbers

Question 100Question →

A weather balloon's altitude in meters after $t$ minutes is given by the function $A(t) = 100 + 50t$ . The air temperature in degrees Celsius at an altitude of $a$ meters is modeled by the function $T(a) = 25 - 0.02a$ . What is the temperature of the air surrounding the balloon, in degrees Celsius, after $2$ minutes?

Show answer & explanation

Answer: 21

Answer

The temperature of the air surrounding the balloon after 2 minutes is 21 degrees Celsius.

The temperature of the air surrounding the balloon after 2 minutes is found by first calculating the balloon's altitude,

A(2) = 100 + 50(2) = 200

meters, and then using this altitude to evaluate the temperature function,

T(200) = 25 - 0.02(200) = 21

.

Step-by-Step Solution

1

Calculate the altitude of the weather balloon at

t = 2

minutes using the function

A(t)

.

A(2) = 200

meters

To find the temperature surrounding the balloon, we must first determine its altitude at the given time of 2 minutes.

2

Substitute the altitude of 200 meters into the temperature function

T(a)

.

T(200) = 21

degrees Celsius

The temperature function depends on the altitude, so evaluating

T(200)

yields the temperature at that height.

Key Concept

Evaluating composite functions in a real-world context.

Answer

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