Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 13

Simplify the following algebraic expression:

\left( \frac{12p^4 q^{-3}}{3p^{-2} q^2} \right)^{-1}

What is the fully simplified form of this expression?

\frac{4q^5}{p^6}

\frac{4p^6}{q^5}

\frac{q^5}{4p^6}

\frac{q}{4p^2}

Show answer & explanation

Answer: $\frac{q^5}{4p^6}$

Answer

The expression simplifies to

\frac{q^5}{4p^6}

.

The correct answer is derived by first simplifying the inner fraction using the quotient rule for exponents, resulting in

4p^6q^{-5}

. Applying the outer negative exponent of

-1

takes the reciprocal of each factor, leading to

4^{-1}p^{-6}q^5

, which is written with positive exponents as

\frac{q^5}{4p^6}

.

Step-by-Step Solution

1

Divide the numerical coefficients.

12 \div 3 = 4

Coefficients follow standard division rules.

2

Simplify the variable

p

using the quotient rule for exponents.

p^4 \div p^{-2} = p^{4 - (-2)} = p^6

When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator.

3

Simplify the variable

q

using the quotient rule for exponents.

q^{-3} \div q^2 = q^{-3 - 2} = q^{-5}

Subtracting a positive exponent is mathematically equivalent to adding its negative value.

4

Combine the simplified parts into a single expression inside the parentheses.

4p^6q^{-5}

This represents the fully simplified value of the fraction before applying the outer exponent.

5

Apply the outer exponent of

-1

to every factor in the expression.

(4p^6q^{-5})^{-1} = 4^{-1} \cdot (p^6)^{-1} \cdot (q^{-5})^{-1}

The power of a product rule states that

(abc)^x = a^x b^x c^x

.

6

Simplify the resulting exponents and write with positive exponents.

4^{-1}p^{-6}q^5 = \frac{q^5}{4p^6}

A negative exponent indicates the reciprocal of the base (

x^{-n} = 1/x^n

).

Key Concept

Simplifying algebraic fractions using the quotient, product, and negative power rules for exponents.

Question 242Question →

Determine the value of the polynomial $-2a^2 + 5ab - b^3$ under the condition that $a = -2$ and $b = -3$ . Which of the following represents the correct evaluation?

-5

65

49

73

Show answer & explanation

Answer: 49

Answer

49

The correct answer is found by substituting the variable values, properly applying the exponents to the negative numbers (where

(-2)^2 = 4

and

(-3)^3 = -27

), and adhering to the order of operations to multiply before adding and subtracting.

Step-by-Step Solution

1

Substitute

-2

for

a

and

-3

for

b

into the given polynomial.

-2(-2)^2 + 5(-2)(-3) - (-3)^3

To prepare the expression for numerical evaluation.

2

Evaluate the exponential terms, noting that a negative number raised to an even power is positive, and to an odd power is negative.

-2(4) + 5(-2)(-3) - (-27)

The order of operations (PEMDAS) requires resolving exponents before multiplication.

3

Perform the multiplication operations from left to right.

-8 + 30 - (-27)

The order of operations dictates that multiplication must be completed before addition and subtraction.

4

Convert the subtraction of a negative into addition and calculate the final sum.

-8 + 30 + 27 = 22 + 27 = 49

This is the final step to find the total value of the polynomial.

Key Concept

Evaluating Algebraic Expressions

Estimated Time:45s

Question 243Question →

When the algebraic expression $(4x - 3)(2x^2 + 5x - 1)$ is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of the $x$ term?

Show answer & explanation

Answer: -19

Answer

-19

The correct answer is

-19

because the

x

term is exclusively formed by the sum of the products

(4x)(-1)

and

(-3)(5x)

, which equals

-4x - 15x = -19x

.

Step-by-Step Solution

1

Identify the goal.

We need to find the coefficient of the

x

term in the product of

(4x - 3)

and

(2x^2 + 5x - 1)

.

Focusing only on the terms that produce an

x

term saves time compared to expanding the entire polynomial expression.

2

Find the first partial product that results in an

x

term.

Multiply the

x

term from the binomial (

4x

) by the constant term from the trinomial (

-1

) to get

-4x

.

An

x

term multiplied by a constant results in an

x

term.

3

Find the second partial product that results in an

x

term.

Multiply the constant term from the binomial (

-3

) by the

x

term from the trinomial (

5x

) to get

-15x

.

A constant multiplied by an

x

term results in an

x

term.

4

Combine the partial products.

-4x + (-15x) = -19x

To find the total

x

term in the expanded polynomial, all like terms must be combined.

5

Identify the final coefficient.

The coefficient of

-19x

is

-19

.

The coefficient is the numerical multiplier of the variable.

Key Concept

Multiplying Polynomials and Combining Like Terms

Question 244Question →

Evaluate the algebraic expression $P = 4v^3 - 3uv + 2u^2$ using the assigned values $u = -4$ and $v = -2$ . Fill in the missing numerical values for the intermediate calculation step and the final evaluated result.

Fill in the blanks below

The calculated value for the isolated term

4v^3

is , which makes the final evaluated total for the entire expression

P

equal to .

Show answer & explanation

Answer

The first term evaluates to -32, and the entire expression evaluates to -24.

The expression requires careful substitution of

u = -4

and

v = -2

. Following the order of operations (PEMDAS), we first calculate the exponents:

(-2)^3 = -8

and

(-4)^2 = 16

. Next, we perform the multiplication for each term:

4(-8) = -32

,

-3(-4)(-2) = -24

, and

2(16) = 32

. Finally, combining these intermediate values gives

-32 - 24 + 32 = -24

.

Step-by-Step Solution

1

Substitute the value of v into the first term

4v^3

and evaluate according to the order of operations.

4(-2)^3 = 4(-8) = -32

Exponents must be evaluated before multiplication. The cube of a negative number remains negative.

2

Substitute the values of u and v into the second term

-3uv

and evaluate.

-3(-4)(-2) = -3(8) = -24

Multiplying two negative numbers yields a positive product, which is then multiplied by the negative coefficient.

3

Substitute the value of u into the third term

2u^2

and evaluate.

2(-4)^2 = 2(16) = 32

The square of a negative number is positive. This must be calculated before multiplying by the coefficient.

4

Combine the evaluated terms to find the final total for P.

-32 - 24 + 32 = -24

Adding and subtracting the values from left to right yields the final result.

Key Concept

Evaluating multi-variable algebraic expressions by correctly substituting negative integers and rigorously applying the order of operations.

Question 245Question →

An algebraic equation requires simplification to isolate the unknown variable. Which of the following represents the correct value of $y$ in the equation below?

3(2y + 5) - y = 2(2y - 1) + 13

0

6

-4

26

Show answer & explanation

Answer: -4

Answer

The correct value for

y

is

-4

.

The correct answer is found by properly applying the distributive property to both sides, combining the resulting like terms, and systematically isolating the variable. Maintaining the balance of the equation by performing inverse operations yields a final value of -4.

Step-by-Step Solution

1

Distribute the constants into the parentheses on both sides of the equation.

6y + 15 - y = 4y - 2 + 13

To eliminate the parentheses and prepare the equation for combining terms.

2

Combine like terms on each individual side.

5y + 15 = 4y + 11

To simplify the expressions on both sides before isolating the variable.

3

Subtract

4y

from both sides.

y + 15 = 11

To group all variable terms onto the left side of the equation.

4

Subtract

15

from both sides.

y = -4

To completely isolate

y

and find the final root of the equation.

Key Concept

Solving one-variable linear equations by applying the distributive property and combining like terms.

Question 246Question →

What value of $x$ satisfies the equation $\frac{1}{2}(4x - 6) + 3x = \frac{2}{3}(9x - 12)$ ?

5

-5

6

11

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. By distributing the fractions, combining the like terms, and properly using inverse operations to isolate

x

, the equation simplifies step-by-step to

x = 5

.

Step-by-Step Solution

1

Distribute the fractions into the parentheses on both sides of the equation.

2x - 3 + 3x = 6x - 8

Applying the distributive property removes the parentheses and prepares the equation for combining terms.

2

Combine the like terms on the left side.

5x - 3 = 6x - 8

Simplifying each side individually makes it easier to isolate the variable.

3

Subtract

5x

from both sides to group the variable terms on the right side.

-3 = x - 8

Keeping the variable coefficient positive reduces the chance of a sign error.

4

Add

8

to both sides to isolate

x

.

5 = x

Applying the inverse operation isolates the variable and provides the final solution.

Key Concept

Solving One-Variable Linear Equations

Question 247Question →

Determine the simplified form of the following product, assuming $c$ and $d$ are non-zero real numbers:

(2c^{-3}d^5)^{-2}(4c^2d^{-1})

What is the resulting fully simplified expression?

-\frac{16c^8}{d^{11}}

c^{12}d^{10}

\frac{c^8}{d^{11}}

\frac{8c^8}{d^{11}}

Show answer & explanation

Answer: $\frac{c^8}{d^{11}}$

Answer

The fully simplified expression is

\frac{c^8}{d^{11}}

.

The correct answer is obtained by first applying the power of a power rule to

(2c^{-3}d^5)^{-2}

, yielding

\frac{1}{4}c^6d^{-10}

. Multiplying this by

4c^2d^{-1}

involves adding the exponents of like bases, which results in

c^8d^{-11}

. Moving the negative exponent to the denominator yields the fully simplified form.

Step-by-Step Solution

1

Apply the power of a product rule to the first term, distributing the

-2

exponent to the coefficient and both variables.

2^{-2}(c^{-3})^{-2}(d^5)^{-2} = \frac{1}{4}c^6d^{-10}

The rules

(xy)^a = x^ay^a

and

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

require every part of the base to be raised to the outer power.

2

Multiply the expanded first term by the second term by grouping the coefficients and the like variables together.

(\frac{1}{4} \cdot 4)(c^6 \cdot c^2)(d^{-10} \cdot d^{-1})

Multiplication is commutative, which allows rearranging the terms to combine matching bases.

3

Multiply the numerical coefficients and add the exponents for the variables with the same base.

1 \cdot c^{6+2} \cdot d^{-10+(-1)} = c^8d^{-11}

The product of powers rule states that when multiplying terms with the same base, their exponents should be added (

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

).

4

Rewrite the expression so that all exponents are positive.

\frac{c^8}{d^{11}}

A fully simplified algebraic expression is typically written with positive exponents, using the rule

x^{-a} = \frac{1}{x^a}

.

Key Concept

Operations with Variable Exponents

Question 248Question →

When the expression $(3x - 4)(2x + 5) - x(x - 2)$ is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of $x$ ?

Show answer & explanation

Answer: 9

Answer

9

Expanding the binomials gives

6x^2 + 7x - 20

. Distributing

-x

gives

-x^2 + 2x

. Combining like terms results in

5x^2 + 9x - 20

. The question asks for the coefficient of

x

, which is the number multiplying the

x

term: 9.

Step-by-Step Solution

1

Multiply the binomials

(3x - 4)

and

(2x + 5)

.

6x^2 + 15x - 8x - 20

, which simplifies to

6x^2 + 7x - 20

.

To expand the first part of the expression using the distributive property (FOIL).

2

Distribute

-x

across the binomial

(x - 2)

.

-x^2 + 2x

.

To expand the second part of the expression, ensuring the negative sign is applied to both terms inside the parentheses.

3

Combine the expanded parts.

(6x^2 + 7x - 20) + (-x^2 + 2x) = 5x^2 + 9x - 20

.

To simplify the full expression into standard polynomial form by grouping like terms.

4

Identify the coefficient of

x

.

The coefficient of

x

in

5x^2 + 9x - 20

is

9

.

This answers the specific question asked.

Key Concept

Multiplying polynomials and combining like terms

Question 249Question →

Consider the algebraic expression $\frac{2x^2 - yz}{x + y}$ . If the variables are assigned the values $x = -3$ , $y = 5$ , and $z = -2$ , what are the intermediate evaluated values for the numerator and denominator, and what is the final result?

Fill in the blanks below

After substituting the values, the evaluated result of the numerator is and the denominator is . Dividing these intermediate values yields a final result of .

Show answer & explanation

Answer

The intermediate numerator evaluates to 28, the denominator to 2, and the final evaluated result is 14.

The correct final value is 14. This is determined by carefully substituting the negative and positive variable values into the numerator and denominator, strictly applying the order of operations to handle exponents and negative signs, and finally dividing the intermediate numerator 28 by the denominator 2.

Step-by-Step Solution

1

Substitute the value of

x

into the first term of the numerator,

2x^2

.

2(-3)^2 = 2(9) = 18

Applying the order of operations, exponentiation must be evaluated before multiplication.

2

Substitute the values of

y

and

z

into the second term of the numerator,

yz

.

(5)(-2) = -10

Multiplying a positive integer by a negative integer results in a negative integer.

3

Calculate the complete numerator by subtracting the second term from the first.

18 - (-10) = 18 + 10 = 28

Subtracting a negative number is mathematically equivalent to adding its positive counterpart.

4

Substitute the values of

x

and

y

to evaluate the denominator,

x + y

.

-3 + 5 = 2

Combining the negative and positive integers.

5

Divide the fully evaluated numerator by the evaluated denominator.

\frac{28}{2} = 14

The final step in evaluating the rational expression.

Key Concept

Evaluating Algebraic Expressions

Question 250Question →

In a mathematical model, the output variable $M$ is defined by the equation $M = 2a^2 - 3ab + b^3$ . What is the computed value of $M$ if the input parameters are set to $a = -3$ and $b = -2$ ?

-8

10

28

-28

Show answer & explanation

Answer: -8

Answer

-8

The correct answer is found by substituting the values and strictly following the order of operations. First, the exponents are evaluated:

(-3)^2 = 9

and

(-2)^3 = -8

. Next, the multiplication is performed:

2(9) = 18

and

-3(-3)(-2) = -18

. Finally, combining these values gives

18 - 18 - 8 = -8

.

Step-by-Step Solution

1

Substitute the values

a = -3

and

b = -2

into the original equation

M = 2a^2 - 3ab + b^3

.

M = 2(-3)^2 - 3(-3)(-2) + (-2)^3

Direct substitution is the first step in evaluating any algebraic expression.

2

Evaluate the terms with exponents according to the order of operations (PEMDAS/BODMAS).

(-3)^2 = 9

and

(-2)^3 = -8

Exponents must be calculated before multiplication. A negative number squared is positive, while a negative number cubed remains negative.

3

Perform the multiplication for each term in the expression.

2(9) = 18

and

-3(-3)(-2) = -18

Multiplication is performed next. For the middle term, multiplying three negative numbers results in a negative product.

4

Combine the resulting values using addition and subtraction.

18 - 18 + (-8) = 0 - 8 = -8

This final arithmetic operation yields the evaluated result of the expression.

Key Concept

Evaluating algebraic expressions involving exponents, negative bases, and strict adherence to the order of operations.

Estimated Time:1m 0s

Question 251Question →

Solve the following linear equation for $w$ :

\frac{1}{3}(12w - 9) + 5 = 2(w - 1) + 8

Enter the exact integer value that satisfies the equation.

Fill in the blanks below

The value of

w

is .

Show answer & explanation

Answer

2

By successfully distributing the factors, combining like terms on each side, and balancing the equation by moving variables and constants to opposite sides, the isolated variable equates to exactly 2.

Step-by-Step Solution

1

Distribute the coefficients to remove parentheses on both sides of the equation.

4w - 3 + 5 = 2w - 2 + 8

Applying the distributive property simplifies the expression and exposes the variable terms.

2

Combine the constant terms on each side.

4w + 2 = 2w + 6

Simplifying each side makes it easier to balance the equation in the next steps.

3

Subtract

2w

from both sides of the equation.

2w + 2 = 6

This groups all the variable terms onto the left side.

4

Subtract

2

from both sides, then divide the result by

2

.

2w = 4 \Rightarrow w = 2

This completely isolates the variable

w

to reveal its final value.

Key Concept

Solving One-Variable Linear Equations

Question 252Question →

What is the numerical value of the algebraic expression $-5x^2 + \frac{12y}{x} - y^2$ when evaluated for $x = -2$ and $y = -3$ ?

29

7

-11

109

Show answer & explanation

Answer: -11

Answer

The correct numerical value of the expression is -11.

Substituting

x = -2

and

y = -3

gives

-5(-2)^2 + \frac{12(-3)}{-2} - (-3)^2

. Evaluating the exponents first results in

-5(4) + \frac{-36}{-2} - 9

. Multiplying and dividing yields

-20 + 18 - 9

. Finally, adding and subtracting gives the correct value of

-11

.

Step-by-Step Solution

1

Substitute the given numeric values into the algebraic expression using parentheses.

-5(-2)^2 + \frac{12(-3)}{-2} - (-3)^2

Using parentheses during substitution prevents sign errors, especially when variables have negative values.

2

Evaluate the terms with exponents.

-5(4) + \frac{12(-3)}{-2} - (9)

According to the order of operations (PEMDAS), exponents must be evaluated before multiplication and division.

3

Perform the multiplication and division operations.

-20 + \frac{-36}{-2} - 9 \Rightarrow -20 + 18 - 9

Multiplication and division are processed from left to right before addition and subtraction.

4

Complete the evaluation by performing addition and subtraction from left to right.

-20 + 18 - 9 = -2 - 9 = -11

Addition and subtraction are the final steps in the order of operations.

Key Concept

Evaluating algebraic expressions requires careful substitution of values using parentheses and strict adherence to the order of operations (PEMDAS).

Question 253Question →

The dimensions of a rectangular box are given by the expressions $5x$ , $(2x - 3)$ , and $(x + 4)$ . The volume of the box is the product of its three dimensions. When the volume is expressed as a fully simplified polynomial in the form $ax^3 + bx^2 + cx$ , determine the value of the coefficient $b$ .

Fill in the blanks below

The value of

b

is .

Show answer & explanation

Answer

25

To find the coefficient

b

, you must expand the product of the three dimensions. First, multiply the binomials using FOIL:

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

. Then, distribute

5x

across the resulting trinomial:

5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x

. Comparing this to the specified polynomial form

ax^3 + bx^2 + cx

, the term with

x^2

is

25x^2

, making the coefficient

b = 25

.

Step-by-Step Solution

1

Set up the volume expression by multiplying the three given dimensions.

V = 5x(2x - 3)(x + 4)

The volume of a rectangular box is calculated as length

\times

width

\times

height.

2

Multiply the two binomials using the FOIL method.

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

It is mathematically sound and often easier to multiply the binomials first before distributing the monomial.

3

Distribute the monomial

5x

to each term in the resulting trinomial.

5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x

To fully expand the polynomial, every term inside the parentheses must be multiplied by

5x

.

4

Identify the value of

b

by matching the result to the given standard form

ax^3 + bx^2 + cx

.

b = 25

The variable

b

represents the coefficient of the

x^2

term in the fully expanded polynomial.

Key Concept

Multiplying Polynomials and the Distributive Property

Question 254Question →

Given the values $x = \frac{1}{2}$ , $y = -4$ , and $z = -2$ , determine the value of the expression $4x^2y - yz^3$ .

Show answer & explanation

Answer: -36

Answer

-36

The correct answer is derived by substituting the variables accurately, applying the exponents (especially noting that a negative base to an odd power remains negative), and strictly following the order of operations to multiply and then subtract.

Step-by-Step Solution

1

Substitute

x = \frac{1}{2}

,

y = -4

, and

z = -2

into the expression.

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

Variable substitution must be done exactly as given to prepare for evaluation.

2

Evaluate the terms with exponents.

(\frac{1}{2})^2 = \frac{1}{4}

and

(-2)^3 = -8

. The expression becomes

4(\frac{1}{4})(-4) - (-4)(-8)

.

According to the order of operations (PEMDAS), exponents must be resolved before multiplication.

3

Perform the multiplication operations from left to right.

4(\frac{1}{4})(-4) = 1(-4) = -4

and

(-4)(-8) = 32

. The expression is now

-4 - 32

.

Multiplication is the next step in the order of operations. Note that multiplying two negative numbers results in a positive product.

4

Perform the final subtraction.

-4 - 32 = -36

Subtracting a positive number from a negative number moves further left on the number line.

Key Concept

Evaluating algebraic expressions using rational numbers, negative bases, and the correct order of operations.

Estimated Time:1m 0s

Question 255Question →

If the polynomial expression $(3x - 5)$ is multiplied by the expression $(Ax^2 + 2x - 7)$ , the resulting simplified polynomial has an $x^2$ coefficient of $-19$ . What is the value of the constant $A$ ?

Show answer & explanation

Answer: 5

Answer

5

The

x^2

terms in the product of

(3x - 5)

and

(Ax^2 + 2x - 7)

are generated by multiplying the

x

terms together

(3x \cdot 2x = 6x^2)

and by multiplying the constant by the

x^2

term

(-5 \cdot Ax^2 = -5Ax^2)

. Setting their sum equal to

-19x^2

gives the equation

6 - 5A = -19

. Solving for

A

yields

A = 5

.

Step-by-Step Solution

1

Identify combinations that yield an

x^2

term

The

x^2

terms come from multiplying

(3x)

by

(2x)

and

(-5)

by

(Ax^2)

.

To find the coefficient of a specific term without expanding the entire polynomial, only multiply the terms whose exponents sum to the target power.

2

Calculate the partial products

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

and

(-5)(Ax^2) = -5Ax^2

These represent the only contributions to the

x^2

term in the final expanded polynomial.

3

Formulate an equation for the coefficient

6 - 5A = -19

The sum of the partial

x^2

coefficients must equal the total

x^2

coefficient given in the problem.

4

Solve the linear equation for

A

-5A = -25 \rightarrow A = 5

Isolating the variable

A

provides the final answer.

Key Concept

Multiplying Polynomials

Question 256Question →

Calculate the result of the following algebraic fraction when $x = -2$ and $y = 2$ .

Fill in the blanks below

The evaluated value of the expression

\frac{4x^3 + 2y^2}{x - y}

is .

Show answer & explanation

Answer

6

The correct answer is 6 because substituting the values

x = -2

and

y = 2

into the expression yields a numerator of

-24

and a denominator of

-4

. Dividing a negative by a negative results in a positive 6.

Step-by-Step Solution

1

Substitute the given values for

x

and

y

into the expression.

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

Variable substitution is the first required step in evaluating any algebraic expression.

2

Evaluate the terms with exponents in the numerator.

(-2)^3 = -8

and

2^2 = 4

According to the order of operations (PEMDAS), exponents must be resolved before multiplication.

3

Perform the multiplication operations in the numerator.

4(-8) = -32

and

2(4) = 8

Multiplication is performed after exponents.

4

Add the resulting terms in the numerator and subtract the terms in the denominator.

Numerator:

-32 + 8 = -24

. Denominator:

-2 - 2 = -4

.

Simplify the numerator and denominator completely before dividing.

5

Divide the simplified numerator by the simplified denominator.

\frac{-24}{-4} = 6

Dividing a negative number by a negative number results in a positive quotient.

Key Concept

Evaluating algebraic expressions with negative integers and exponents

Estimated Time:1m 0s

Question 257Question →

A polynomial $P(x)$ is defined as the product of the binomial $(4x - 3)$ and the trinomial $(-2x^2 + 5x - 1)$ . What is the fully expanded and simplified form of $P(x)$ ?

-8x^3 + 14x^2 - 19x - 3

-2x^2 + x + 3

-8x^3 + 26x^2 - 19x + 3

-8x^3 - 15x + 3

Show answer & explanation

Answer: $-8x^3 + 26x^2 - 19x + 3$

Answer

-8x^3 + 26x^2 - 19x + 3

The correct expression is found by successfully distributing both

4x

and

-3

across the trinomial

(-2x^2 + 5x - 1)

, maintaining correct exponents and signs, and finally combining the like terms:

20x^2

with

6x^2

, and

-4x

with

-15x

.

Step-by-Step Solution

1

Set up the multiplication using the distributive property.

4x(-2x^2 + 5x - 1) - 3(-2x^2 + 5x - 1)

Every term in the first polynomial must be multiplied by every term in the second polynomial.

2

Distribute

4x

through the trinomial.

4x(-2x^2) + 4x(5x) + 4x(-1) = -8x^3 + 20x^2 - 4x

Multiply coefficients and add exponents for the variable base.

3

Distribute

-3

through the trinomial.

-3(-2x^2) - 3(5x) - 3(-1) = 6x^2 - 15x + 3

Multiply the constant by each term, carefully applying integer sign rules.

4

Combine the results and group like terms.

-8x^3 + (20x^2 + 6x^2) + (-4x - 15x) + 3

Grouping terms with the same variable exponent prepares the expression for final simplification.

5

Add the coefficients of the like terms.

-8x^3 + 26x^2 - 19x + 3

This yields the polynomial in standard form.

Key Concept

Multiplying polynomials by distributing every term in the first factor to every term in the second, then combining like terms.

Question 258Question →

Evaluate the given algebraic equation to solve for $p$ :

7 - 3(p + 2) = p - 7

Which of the following represents the correct solution for $p$ ?

5

-5

-4

2

Show answer & explanation

Answer: 2

Answer

The correct value is 2.

The correct approach involves distributing the

-3

into the parentheses to get

7 - 3p - 6

, combining like terms to get

1 - 3p

, and then balancing the equation to solve for

p

. Adding

3p

to both sides yields

1 = 4p - 7

, and adding

7

gives

8 = 4p

, which simplifies to

p = 2

.

Step-by-Step Solution

1

Apply the distributive property to the left side of the equation.

7 - 3p - 6 = p - 7

The -3 must be multiplied by both

p

and

2

to eliminate the parentheses.

2

Combine the constant terms on the left side.

1 - 3p = p - 7

Combining

7

and

-6

simplifies the equation before moving terms across the equals sign.

3

Add

3p

to both sides of the equation.

1 = 4p - 7

This groups all terms containing the variable

p

on the right side.

4

Add

7

to both sides and divide by

4

.

8 = 4p \Rightarrow p = 2

This isolates the variable

p

to find the final numerical solution.

Key Concept

Solving Linear Equations with Distribution

Question 259Question →

A manufacturer is cutting a square opening out of a rectangular metal panel. The rectangular panel has a length of $(4x - 1)$ inches and a width of $(2x + 3)$ inches. The square opening to be removed has a side length of $(x + 2)$ inches. Which of the following expressions represents the total remaining area of the metal panel after the square is removed?

7x^2 + 6x - 7

7x^2 + 10x - 7

7x^2 + 14x + 1

7x^2 - 7

Show answer & explanation

Answer: $7x^2 + 6x - 7$

Answer

The correct expression is

7x^2 + 6x - 7

.

The remaining area is found by subtracting the square's area from the rectangle's area. Multiplying the dimensions

(4x - 1)(2x + 3)

yields

8x^2 + 10x - 3

. Squaring the dimension

(x + 2)

yields

x^2 + 4x + 4

. Subtracting the entire second polynomial requires distributing the negative sign, resulting in

8x^2 + 10x - 3 - x^2 - 4x - 4

, which correctly simplifies to

7x^2 + 6x - 7

.

Step-by-Step Solution

1

Calculate the area of the rectangular panel.

(4x - 1)(2x + 3) = 8x^2 + 12x - 2x - 3 = 8x^2 + 10x - 3

The area of a rectangle is length multiplied by width, which requires multiplying two binomials using the FOIL method.

2

Calculate the area of the square opening.

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

The area of a square is its side length multiplied by itself.

3

Subtract the square's area from the rectangle's area.

(8x^2 + 10x - 3) - (x^2 + 4x + 4) = 8x^2 + 10x - 3 - x^2 - 4x - 4 = 7x^2 + 6x - 7

To find the remaining area, the removed portion must be subtracted. It is critical to distribute the negative sign to every term inside the parentheses being subtracted.

Key Concept

Multiplying binomials and subtracting polynomials while properly distributing negative signs.

Question 260Question →

Which of the following represents the value of $y$ that makes the given equation true?

\frac{2y - 5}{3} + 8 = 3

-5

0

5

19

Show answer & explanation

Answer: -5

Answer

The correct value for the variable is -5.

The solution requires a sequence of inverse operations to isolate the variable. First, subtracting 8 from both sides gives a value of -5 on the right. Then, multiplying by 3 clears the fraction, resulting in 2y - 5 = -15. Adding 5 yields 2y = -10, and dividing by 2 confirms the final value is -5.

Step-by-Step Solution

1

Isolate the fractional term by subtracting 8 from both sides of the equation.

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

Applying the inverse operation of addition moves the constant term to the right side, simplifying the equation.

2

Multiply both sides of the equation by 3.

2y - 5 = -15

Multiplying by the denominator eliminates the fraction, making it possible to solve for the variable.

3

Add 5 to both sides of the equation.

2y = -10

This step isolates the term with the variable on the left side.

4

Divide both sides by 2.

y = -5

Dividing by the coefficient of the variable gives the final exact value.

Key Concept

Solving one-variable linear equations involving fractions

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