Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

What is the completely factored form of the polynomial $3x^2 - 12x - 36$ ?

3(x - 6)(x + 2)

3(x + 6)(x - 2)

(3x + 6)(x - 6)

3(x - 6)(x - 2)

Show answer & explanation

Answer: $3(x - 6)(x + 2)$

Answer

3(x - 6)(x + 2)

To completely factor the polynomial, the greatest common factor must be extracted first. Factoring out the GCF of 3 yields

3(x^2 - 4x - 12)

. Next, factoring the trinomial

x^2 - 4x - 12

requires two integers that multiply to

-12

and sum to

-4

. These integers are

-6

and

2

. Therefore, the completely factored form is

3(x - 6)(x + 2)

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all terms in the polynomial.

3(x^2 - 4x - 12)

Extracting the GCF simplifies the expression, making the quadratic trinomial easier to factor.

2

Find two numbers that multiply to the constant term (

-12

) and add to the coefficient of the middle term (

-4

).

The numbers are

-6

and

2

, because

(-6) \times 2 = -12

and

(-6) + 2 = -4

.

These values are required to correctly split the middle term and factor the quadratic trinomial.

3

Write the completely factored form by combining the GCF and the two binomial factors.

3(x - 6)(x + 2)

This represents the polynomial broken down into its simplest, irreducible algebraic factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Question 22Question →

Complete the factored form of the given quadratic polynomial by determining the missing constant.

What number correctly fills in the blank?

Fill in the blanks below

$n^2 + 11n + 24 = (n + 3)(n + )$

Show answer & explanation

Answer

8

The polynomial

n^2 + 11n + 24

factors into two binomials. We are given one factor,

(n + 3)

. To find the missing value in the second factor

(n + b)

, we can use the property that the product of the constant terms in the binomials must equal the constant term of the trinomial. Therefore,

3 \times b = 24

, which means

b = 8

. We verify this by adding the constants:

3 + 8 = 11

, which matches the coefficient of the middle term. Thus, the correct value for the blank is 8.

Step-by-Step Solution

1

Identify the relationship between the constant term of the polynomial and the constants in its binomial factors.

The product of the constants in the binomial factors must equal the constant term of the polynomial, 24.

When expanding

(n + a)(n + b)

, the constant term is calculated as

a \times b

.

2

Set up an equation to find the missing value.

3 \times \text{missing value} = 24

.

One of the binomial factors is given as

(n + 3)

, so one constant is 3.

3

Solve for the missing value and verify with the middle term.

The missing value is 8, since

24 \div 3 = 8

. Checking the middle term,

3 + 8 = 11

, which matches

11n

.

This confirms that 8 satisfies both the multiplication to 24 and the addition to 11 requirements for factoring the trinomial.

Key Concept

Factoring quadratic trinomials of the form x^2 + bx + c.

Question 23Question →

Connect the unsimplified exponential expressions in the left column to their corresponding reduced forms in the right column, assuming $x$ and $y$ are non-zero constants.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each unsimplified expression matches to a unique reduced form by carefully applying the power, product, and quotient rules of exponents.

Each expression is correctly simplified by systematically applying the product, quotient, and power rules of exponents, paying special attention to numerical coefficients and negative signs.

Step-by-Step Solution

1

Simplify the first expression by expanding the numerator's power and applying the quotient rule.

\frac{(2x^{-3}y^2)^3}{4x^{-5}y^{-1}} = \frac{8x^{-9}y^6}{4x^{-5}y^{-1}} = 2x^{-4}y^7 = \frac{2y^7}{x^4}

The power of a product rule applies exponents to coefficients, while the quotient rule requires subtracting denominator exponents from numerator exponents.

2

Simplify the second expression by resolving the negative outer exponent and multiplying the coefficients and variables.

(3x^4y^{-2})^{-2} \cdot (9x^0y^5) = (3^{-2}x^{-8}y^4) \cdot (9 \cdot 1 \cdot y^5) = \frac{1}{9}x^{-8}y^4 \cdot 9y^5 = x^{-8}y^9 = \frac{y^9}{x^8}

Applying negative exponents to the coefficient creates a fraction, and recognizing that

x^0 = 1

simplifies the second term before multiplication.

3

Simplify the third expression by resolving the fraction inside first, then applying the external negative exponent.

\left( \frac{15x^3y^{-4}}{5x^{-2}y^3} \right)^{-1} = (3x^5y^{-7})^{-1} = 3^{-1}x^{-5}y^7 = \frac{y^7}{3x^5}

Simplifying the internal expression first reduces algebraic errors before applying the reciprocal operation for the -1 exponent.

4

Simplify the fourth expression by multiplying the numerator terms, then dividing by the denominator.

\frac{6x^{-1}y^4 \cdot 2x^5y^{-2}}{3x^8y^{-3}} = \frac{12x^4y^2}{3x^8y^{-3}} = 4x^{4-8}y^{2-(-3)} = 4x^{-4}y^5 = \frac{4y^5}{x^4}

Combining like terms in the numerator through multiplication organizes the expression for a straightforward application of the quotient rule.

Key Concept

Operations with Variable Exponents

Question 24Question →

Which of the following represents the fully simplified form of the algebraic expression below, assuming $x \neq 0$ and $y \neq 0$ ?

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

\frac{y^{10}}{4x^{10}}

\frac{4x^2}{y^6}

\frac{4x^{10}}{y^{10}}

4x^{12}y^8

Show answer & explanation

Answer: $\frac{4x^{10}}{y^{10}}$

Answer

The expression simplifies to

\frac{4x^{10}}{y^{10}}

.

The correct answer is derived by first applying the quotient rule to simplify the expression inside the parentheses, being careful to subtract negative exponents correctly (e.g.,

y^{4 - (-1)} = y^5

). Then, the negative exponent on the outside of the parentheses means taking the reciprocal of the fraction and squaring all terms, which applies to the coefficient (2 becomes 4) and the variables (

x^5

becomes

x^{10}

).

Step-by-Step Solution

1

Simplify the numerical coefficients inside the parentheses.

\frac{2}{4} = \frac{1}{2}

Simplifying the fraction before handling variables reduces computational complexity.

2

Apply the quotient rule for exponents to the variables

x

and

y

.

For

x

:

-3 - 2 = -5

. For

y

:

4 - (-1) = 4 + 1 = 5

. The expression inside the parentheses becomes

\frac{1}{2} x^{-5}y^5

or

\frac{y^5}{2x^5}

.

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

3

Rewrite the entire expression with the simplified inner term.

\left( \frac{y^5}{2x^5} \right)^{-2}

This establishes the simplified base expression before applying the final outer exponent.

4

Apply the negative exponent property to the fraction.

\left( \frac{2x^5}{y^5} \right)^2

A negative exponent on a fraction indicates the reciprocal of that fraction raised to the positive exponent.

5

Apply the power of a quotient and product rules.

\frac{2^2(x^5)^2}{(y^5)^2} = \frac{4x^{10}}{y^{10}}

When raising a power to a power, multiply the exponents, and ensure the outer exponent is applied to the numerical coefficient.

Key Concept

Operations with Variable Exponents

Estimated Time:2m 0s

Question 25Question →

The area of a rectangular board is given by the algebraic expression $4x^2 - 8x - 5$ . Which of the following shows the dimensions of the board expressed as a product of two binomials?

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

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

(4x + 1)(x - 5)

(2x - 1)(2x - 5)

Show answer & explanation

Answer: $(2x + 1)(2x - 5)$

Answer

The correct dimensions are given by the factored form

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

.

To factor

4x^2 - 8x - 5

, find two values that multiply to

-20

(which is

4 \times -5

) and add to

-8

. These values are

-10

and

2

. Splitting the middle term gives

4x^2 - 10x + 2x - 5

. Factoring by grouping gives

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

, which then factors out to

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

.

Step-by-Step Solution

1

Identify the coefficients for the quadratic expression

ax^2 + bx + c

.

a = 4

,

b = -8

, and

c = -5

.

To factor a non-monic quadratic trinomial by grouping, we first need to determine the product of

a

and

c

.

2

Find two numbers that multiply to the product of

a

and

c

(

-20

) and add to

b

(

-8

).

The two numbers are

-10

and

2

.

These numbers allow us to split the middle term (

-8x

) into two parts that can be factored by grouping.

3

Rewrite the original expression by splitting the middle term.

4x^2 - 10x + 2x - 5

This sets up the expression for factoring by grouping.

4

Factor the greatest common factor (GCF) from the first two terms and the last two terms.

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

Extracting the GCF reveals a common binomial factor in both groups.

5

Factor out the common binomial factor.

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

This provides the completely factored form of the original area expression.

Key Concept

Factoring Non-Monic Quadratic Trinomials

Question 26Question →

Simplify the following algebraic expression completely. Assume all variables represent non-zero real numbers.

\left( \frac{-3 x^{-2} y^5}{z^{-3}} \right)^{-2} \times \left( \frac{6 x^4 z^{-1}}{y^{-3}} \right)^3

Which of the following represents the fully simplified expression?

1944 x^8 y^{19} z^3

24 x^3 y^3 z^7

\frac{24 x^{16}}{y z^9}

-\frac{18 x^{16}}{y z^9}

Show answer & explanation

Answer: $\frac{24 x^{16}}{y z^9}$

Answer

The fully simplified expression is

\frac{24 x^{16}}{y z^9}

.

The correct answer is found by first distributing the outer exponents to all terms inside the parentheses (including coefficients), which yields (1/9) * x^4 * y^(-10) * z^(-6) and 216 * x^12 * z^(-3) * y^9. Multiplying these expressions by combining the coefficients and adding the exponents of like variables gives 24 * x^16 * y^(-1) * z^(-9). Rewriting with positive exponents moves y and z to the denominator, resulting in the correct simplified fraction.

Step-by-Step Solution

1

Resolve negative exponents in the denominators of both fractions to bring the variables to the numerators.

\left( -3 x^{-2} y^5 z^3 \right)^{-2} \times \left( 6 x^4 z^{-1} y^3 \right)^3

Moving negative exponents from the denominator to the numerator simplifies the terms before applying the outer powers.

2

Apply the power of a product rule to the first expression, multiplying each inner exponent by the outer exponent of -2.

(-3)^{-2} x^{(-2)(-2)} y^{(5)(-2)} z^{(3)(-2)} = \frac{1}{9} x^4 y^{-10} z^{-6}

The power of a product rule states that (abc)^x = a^x b^x c^x, and the power of a power rule states that (a^x)^y = a^(xy).

3

Apply the power of a product rule to the second expression, multiplying each inner exponent by the outer exponent of 3.

6^3 x^{(4)(3)} z^{(-1)(3)} y^{(3)(3)} = 216 x^{12} z^{-3} y^9

Distributing the cube to the coefficient and each variable's exponent prepares the expression for multiplication.

4

Multiply the resulting expressions together by grouping coefficients and like bases.

\left( \frac{1}{9} \times 216 \right) \left( x^4 \cdot x^{12} \right) \left( y^{-10} \cdot y^9 \right) \left( z^{-6} \cdot z^{-3} \right)

The commutative property of multiplication allows us to group similar terms to apply the product of powers rule.

5

Simplify the groups by multiplying coefficients and adding the exponents of like bases.

24 x^{16} y^{-1} z^{-9}

The product of powers rule states that a^x * a^y = a^(x+y).

6

Rewrite the expression with only positive exponents.

\frac{24 x^{16}}{y z^9}

Standard simplified form requires all exponents to be positive, so terms with negative exponents are moved to the denominator.

Key Concept

Applying the power of a product, power of a power, quotient of powers, and negative exponent rules to simplify complex algebraic expressions.

Question 27Question →

What is the completely factored form of the polynomial $3x^2 - 3x - 36$ ?

3(x - 4)(x + 3)

3(x + 4)(x - 3)

(3x - 12)(x + 3)

3(x - 6)(x + 2)

Show answer & explanation

Answer: $3(x - 4)(x + 3)$

Answer

The completely factored form is

3(x - 4)(x + 3)

.

The correct answer accurately extracts the greatest common factor of

3

first, and then properly factors the resulting quadratic

x^2 - x - 12

into

(x - 4)(x + 3)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all terms in the polynomial

3x^2 - 3x - 36

.

3(x^2 - x - 12)

Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor.

2

Find two numbers that multiply to the constant term (

-12

) and add up to the coefficient of the middle term (

-1

).

The numbers are

-4

and

3

, because

-4 \times 3 = -12

and

-4 + 3 = -1

.

These numbers are required to construct the binomial factors of the quadratic expression.

3

Write the final completely factored expression by combining the GCF and the two binomial factors.

3(x - 4)(x + 3)

This represents the full decomposition of the original polynomial into its simplest multiplicative components.

Key Concept

Factoring out a greatest common factor (GCF) followed by factoring a basic quadratic trinomial.

Question 28Question →

Match each polynomial expression with its completely factored form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each polynomial with its fully factored equivalent using difference of squares, greatest common factor, and basic trinomial factoring rules.

Each correct pairing applies fundamental algebraic factoring rules: recognizing special binomial products like the difference of squares, pulling out the greatest common factor (GCF) from terms, and finding appropriate numerical binomial factors for quadratic trinomials.

Step-by-Step Solution

1

Analyze each polynomial to determine the applicable factoring method.

Identified difference of squares, GCF, and two basic quadratic trinomials.

Different polynomial structures require specific factoring techniques.

2

Factor the difference of squares:

x^2 - 25

.

Matched with

(x - 5)(x + 5)

.

The pattern

a^2 - b^2

factors into

(a - b)(a + b)

.

3

Factor out the greatest common factor for

4x^2 + 8x

.

Matched with

4x(x + 2)

.

Both terms share a maximum common factor of

4x

.

4

Factor the trinomial

x^2 + 7x + 10

.

Matched with

(x + 2)(x + 5)

.

The numbers

2

and

5

multiply to positive

10

and add to positive

7

.

5

Factor the trinomial

x^2 - 3x - 10

.

Matched with

(x - 5)(x + 2)

.

The numbers

-5

and

2

multiply to negative

10

and add to negative

3

.

Key Concept

Factoring Polynomials

Question 29Question →

A civil engineer is designing a rectangular water reservoir. The length of the reservoir is represented by the polynomial $(4x - 3)$ meters, and the cross-sectional area of the side wall is represented by $(2x^2 + 5x - 1)$ square meters. If the total volume of the reservoir is calculated by multiplying its length by its cross-sectional area, which of the following polynomials represents the total volume in cubic meters?

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

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

8x^3 + 20x^2 - 4x - 3

8x^3 + 16x^2 - 21x + 3

Show answer & explanation

Answer: $8x^3 + 14x^2 - 19x + 3$

Answer

The correct polynomial representing the volume is

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

.

The correct expression is found by properly distributing both terms of the binomial

(4x - 3)

to all three terms of the trinomial

(2x^2 + 5x - 1)

, maintaining correct signs, and accurately combining the resulting like terms.

Step-by-Step Solution

1

Set up the multiplication of the two polynomials.

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

The problem states that the volume is calculated by multiplying the length by the cross-sectional area.

2

Distribute the first term of the binomial,

4x

, to every term in the trinomial.

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

The distributive property requires multiplying every term in the second polynomial by the first term of the first polynomial.

3

Distribute the second term of the binomial,

-3

, to every term in the trinomial.

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

Carefully apply the negative sign to all terms during multiplication, noting that a negative times a negative yields a positive.

4

Combine the results and group the like terms.

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

Like terms (terms with the exact same variable configuration) must be grouped together to simplify the expression.

5

Perform the final addition and subtraction to simplify completely.

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

This yields the fully simplified polynomial in standard form.

Key Concept

Multiplying a binomial by a trinomial using the Distributive Property

Question 30Question →

Simplify the following expression completely. Assume $x$ and $y$ are non-zero real numbers.

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

If the simplified expression is written in the form $Ax^B y^C$ , where $A$ , $B$ , and $C$ are integers, what is the value of $A + B + C$ ?

Show answer & explanation

Answer: -3

Answer

-3

The correct answer is found by systematically applying the laws of exponents. First, distribute the outer exponents to all factors inside the parentheses, being careful with signs and coefficients. The numerator simplifies to

(\frac{1}{9}x^{-4}y^6) \cdot (-27x^{-6}y^6) = -3x^{-10}y^{12}

. The denominator simplifies to

x^8y^{-6}

. Dividing the numerator by the denominator using the quotient rule yields

-3x^{-18}y^{18}

. Setting this equal to the target form

Ax^By^C

yields

A = -3

,

B = -18

, and

C = 18

. The sum

A + B + C

is

-3 - 18 + 18 = -3

.

Step-by-Step Solution

1

Apply the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

(a^m)^n = a^{m \cdot n}

to the first factor in the numerator.

(3x^2y^{-3})^{-2} = 3^{-2} \cdot x^{2(-2)} \cdot y^{-3(-2)} = \frac{1}{9}x^{-4}y^6

To eliminate the outer negative exponent on the first term.

2

Apply the same exponent rules to the second factor in the numerator.

(-3x^{-2}y^2)^3 = (-3)^3 \cdot x^{-2(3)} \cdot y^{2(3)} = -27x^{-6}y^6

To eliminate the outer exponent on the second term.

3

Multiply the simplified factors in the numerator using the product rule

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

.

(\frac{1}{9}x^{-4}y^6) \cdot (-27x^{-6}y^6) = (\frac{1}{9} \cdot -27) \cdot x^{-4-6} \cdot y^{6+6} = -3x^{-10}y^{12}

To combine the expressions in the numerator into a single term.

4

Apply the power rules to simplify the denominator.

(x^{-4}y^3)^{-2} = x^{-4(-2)} \cdot y^{3(-2)} = x^8y^{-6}

To eliminate the outer negative exponent in the denominator.

5

Divide the numerator by the denominator using the quotient rule

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

.

\frac{-3x^{-10}y^{12}}{x^8y^{-6}} = -3 \cdot x^{-10-8} \cdot y^{12-(-6)} = -3x^{-18}y^{18}

To reach the fully simplified algebraic form.

6

Identify the values of

A

,

B

, and

C

to calculate the required sum.

A = -3

,

B = -18

,

C = 18

. The sum is

-3 + (-18) + 18 = -3

.

To answer the specific question asked in the stem.

Key Concept

Applying multiple rules of exponents (power of a power, negative exponents, product and quotient rules) to simplify complex algebraic expressions.

Question 31Question →

Which of the following represents the completely factored form of the algebraic expression $3x^2 - 27$ ?

3(x - 3)(x + 3)

3(x^2 - 9)

3(x - 3)(x - 3)

3x(x - 9)

Show answer & explanation

Answer: $3(x - 3)(x + 3)$

Answer

The completely factored form is

3(x - 3)(x + 3)

.

The expression is completely factored by first extracting the greatest common factor of 3 to get

3(x^2 - 9)

, and then factoring the resulting difference of squares into

(x - 3)(x + 3)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all terms in the expression.

3(x^2 - 9)

Both terms share a common factor of 3. Factoring this out simplifies the remaining polynomial.

2

Examine the expression inside the parentheses to see if it can be factored further.

x^2 - 9

is a difference of squares.

The expression

x^2 - 9

follows the pattern

a^2 - b^2

, where

a = x

and

b = 3

.

3

Factor the difference of squares using the standard algebraic pattern

(a - b)(a + b)

.

(x - 3)(x + 3)

Expanding

(x - 3)(x + 3)

yields

x^2 + 3x - 3x - 9

, which simplifies back to

x^2 - 9

.

4

Combine the GCF and the factored binomials to write the final completely factored form.

3(x - 3)(x + 3)

An expression is only completely factored when all of its factors are prime (cannot be factored further).

Key Concept

Factoring by finding a Greatest Common Factor (GCF) followed by factoring a difference of squares.

Question 32Question →

During the mathematical modeling of an electromagnetic field, an engineer derives the formula

\frac{(4a^3b^{-2})^{-2}}{2a^{-4}b^5}

to represent the field intensity. Which of the following expressions is mathematically equivalent to this derived formula?

\frac{2}{a^2b}

\frac{a^5}{32b^9}

\frac{1}{32a^2b}

\frac{1}{32a^{10}b}

Show answer & explanation

Answer: $\frac{1}{32a^2b}$

Answer

The mathematically equivalent expression is

\frac{1}{32a^2b}

.

To find the equivalent expression, first apply the power of a power rule to the numerator. Distribute the exponent -2 to each factor: 4^(-2) * (a^3)^(-2) * (b^-2)^(-2), which simplifies to (1/16) * a^-6 * b^4. Next, divide this result by the denominator, 2a^-4 * b^5. Divide the coefficients: (1/16) / 2 = 1/32. Divide the variables by subtracting the exponents of like bases: for a, it is -6 - (-4) = -2, and for b, it is 4 - 5 = -1. This yields (1/32) * a^-2 * b^-1. Finally, move the variables with negative exponents to the denominator to attain the fully simplified form.

Step-by-Step Solution

1

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

(4)^{-2}(a^3)^{-2}(b^{-2})^{-2} = \frac{1}{16}a^{-6}b^4

When an expression in parentheses is raised to a power, the outer exponent must be applied to the numerical coefficient and multiplied by the exponent of each variable inside.

2

Divide the coefficients of the numerator and the denominator.

\frac{1/16}{2} = \frac{1}{16} \times \frac{1}{2} = \frac{1}{32}

The constant factors can be evaluated and divided separately from the variable terms.

3

Apply the quotient of powers rule for the variable bases.

a^{-6 - (-4)} = a^{-2}

and

b^{4 - 5} = b^{-1}

When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

4

Rewrite the expression using only positive exponents.

\frac{1}{32}a^{-2}b^{-1} = \frac{1}{32a^2b}

Final simplified algebraic expressions are standardly written by moving variables with negative exponents to the denominator.

Key Concept

Applying multiple properties of exponents, including the power of a product rule, power of a power rule, and quotient of powers rule with negative integers.

Question 33Question →

During an analysis of a parabolic trajectory, an engineer must factor the polynomial $4x^2 - 4x - 48$ completely to determine its zero points. What is the correctly factored form of this polynomial?

4(x - 4)(x + 3)

4(x + 4)(x - 3)

(4x - 16)(x + 3)

4(x - 6)(x + 2)

Show answer & explanation

Answer: $4(x - 4)(x + 3)$

Answer

The completely factored form of the polynomial is

4(x - 4)(x + 3)

.

To factor completely, first factor out the GCF of 4 to get

4(x^2 - x - 12)

. Then, factor the resulting trinomial by finding two numbers that multiply to -12 and add to -1. Those numbers are -4 and 3, yielding the final form

4(x - 4)(x + 3)

.

Step-by-Step Solution

1

Extract the greatest common factor (GCF) from all terms in the polynomial

4x^2 - 4x - 48

.

4(x^2 - x - 12)

Factoring out the GCF simplifies the polynomial and makes the remaining quadratic expression easier to factor.

2

Find two integers that multiply to the constant term (-12) and add up to the coefficient of the middle term (-1).

The two integers are -4 and +3.

These integers provide the constants needed for the two binomial factors of the quadratic trinomial.

3

Write the final expression by combining the GCF with the factored binomials.

4(x - 4)(x + 3)

This shows the polynomial broken down into its prime, indivisible factors.

Key Concept

Factoring out a Greatest Common Factor (GCF) before factoring a quadratic trinomial.

Question 34Question →

When the algebraic expression $12y^3 + 12y^2 - 45y$ is factored completely, which of the following expressions is produced?

3y(2y - 3)(2y + 5)

3y(2y + 3)(2y - 5)

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

-3y(2y - 3)(2y + 5)

Show answer & explanation

Answer: $3y(2y - 3)(2y + 5)$

Answer

The completely factored form is

3y(2y - 3)(2y + 5)

.

The correct answer accurately extracts the full GCF of

3y

and correctly factors the remaining trinomial into

(2y - 3)(2y + 5)

. Expanding this result confirms it equals

12y^3 + 12y^2 - 45y

.

Step-by-Step Solution

1

Identify and extract the Greatest Common Factor (GCF) from all terms in

12y^3 + 12y^2 - 45y

.

The GCF is

3y

. Factoring it out leaves

3y(4y^2 + 4y - 15)

.

Extracting the GCF first simplifies the polynomial and is a required step for complete factorization.

2

Factor the remaining non-monic trinomial,

4y^2 + 4y - 15

, by finding two numbers that multiply to

a \times c

and add to

b

.

We need numbers that multiply to

(4)(-15) = -60

and add to

4

. The numbers are

10

and

-6

.

This prepares the trinomial for factoring by grouping.

3

Rewrite the middle term (

4y

) using these two numbers and factor by grouping.

4y^2 + 10y - 6y - 15 = 2y(2y + 5) - 3(2y + 5) = (2y - 3)(2y + 5)

.

Grouping extracts common binomial factors, breaking down the quadratic into its simplest linear binomials.

4

Combine the original GCF with the newly factored binomials.

3y(2y - 3)(2y + 5)

To represent the full and complete factorization of the original polynomial.

Key Concept

Factoring polynomials by first extracting the GCF and subsequently using the ac-method (or grouping) to factor a non-monic quadratic trinomial.

Question 35Question →

Simplify the following polynomial expression completely:

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

Which of the following represents the fully expanded and simplified result?

5x^3 - x^2y + 5xy^2 - 8y^3

5x^3 - 13x^2y + 23xy^2 - 8y^3

5x^3 - 7x^2y + 23xy^2 - 8y^3

7x^3 - 13x^2y + 23xy^2 - 8y^3

Show answer & explanation

Answer: $5x^3 - x^2y + 5xy^2 - 8y^3$

Answer

The completely simplified expression is

5x^3 - x^2y + 5xy^2 - 8y^3

.

The correct answer is derived by carefully following the order of operations: first expanding both the left

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

and right

x(x - 3y)^2

sides of the subtraction independently, and then combining the resulting polynomials. Distributing the negative sign correctly during the final subtraction yields

5x^3 - x^2y + 5xy^2 - 8y^3

.

Step-by-Step Solution

1

Expand the product of the binomial and the trinomial:

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

.

6x^3 - 3x^2y + 12xy^2 - 4x^2y + 2xy^2 - 8y^3

Every term in the first binomial must be multiplied by every term in the trinomial to ensure full expansion.

2

Combine the like terms in the first expanded expression.

6x^3 - 7x^2y + 14xy^2 - 8y^3

Combining like terms simplifies the polynomial before performing the final subtraction operation.

3

Expand the squared binomial in the second part of the expression:

(x - 3y)^2

.

x^2 - 6xy + 9y^2

A perfect square binomial

(a-b)^2

expands to

a^2 - 2ab + b^2

.

4

Distribute the

x

into the expanded squared binomial:

x(x^2 - 6xy + 9y^2)

.

x^3 - 6x^2y + 9xy^2

The outer monomial must be distributed to each term in the trinomial.

5

Subtract the second result from the first result:

(6x^3 - 7x^2y + 14xy^2 - 8y^3) - (x^3 - 6x^2y + 9xy^2)

.

6x^3 - 7x^2y + 14xy^2 - 8y^3 - x^3 + 6x^2y - 9xy^2

Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses.

6

Combine all remaining like terms.

5x^3 - x^2y + 5xy^2 - 8y^3

Grouping

x^3

,

x^2y

, and

xy^2

terms produces the final simplified answer in standard form.

Key Concept

Multiplying complex polynomials and applying subtraction across multiple terms

Question 36Question →

The polynomial $x^2 + 6x - 16$ can be factored as $(x - 2)(x + k)$ . What is the value of the constant $k$ ?

Show answer & explanation

Answer: 8; +8; eight

Answer

8

The constant

k

must be 8 because in the factored form

(x - 2)(x + k)

, the product of -2 and

k

must equal the constant term of the polynomial, which is -16. Since

-2 \times 8 = -16

and the sum

-2 + 8 = 6

gives the correct middle coefficient, the value of

k

is exactly 8.

Step-by-Step Solution

1

Analyze the relationship between the expanded polynomial and its factored form.

The expanded form is

x^2 + 6x - 16

and the factored form is

(x - 2)(x + k)

.

This sets up the equation to solve for the missing constant

k

.

2

Use the property of multiplying binomials to find the constant term.

The product of the constants in the binomials must equal the constant term of the polynomial:

-2 \times k = -16

.

When expanding

(x + a)(x + b)

, the constant term is always equal to

a \times b

.

3

Solve the equation for

k

.

k = -16 \div (-2) = 8

.

Dividing both sides by -2 isolates

k

.

4

Verify the middle term coefficient with the found value of

k

.

-2 + 8 = 6

, which matches the middle term

6x

.

Checking the sum of the constants ensures the entire factorization is correct.

Key Concept

Factoring quadratic trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient.

Question 37Question →

When the algebraic expression $5x^2 - 13x - 6$ is factored completely, it can be written in the form $(5x + a)(x - b)$ , where $a$ and $b$ are positive integers. What is the value of $a + b$ ?

Show answer & explanation

Answer: 5; five

Answer

5

Factoring the quadratic

5x^2 - 13x - 6

yields

(5x + 2)(x - 3)

. By comparing this to the target format

(5x + a)(x - b)

, we can identify that

a = 2

and

b = 3

. Adding these together gives the final answer of 5.

Step-by-Step Solution

1

Determine the factoring method for the non-monic quadratic expression

5x^2 - 13x - 6

.

Since the leading coefficient is 5, look for two numbers that multiply to

(5)(-6) = -30

and add to the middle coefficient,

-13

.

This sets up the factor-by-grouping method.

2

Find the two numbers.

The numbers are

-15

and

2

.

(-15) \times 2 = -30

and

-15 + 2 = -13

.

3

Rewrite the middle term using these two numbers and factor by grouping.

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

.

Grouping common terms extracts the binomial factors.

4

Match the factored form to the given format

(5x + a)(x - b)

.

Comparing

(5x + 2)(x - 3)

to

(5x + a)(x - b)

reveals that

a = 2

and

b = 3

.

Both

a

and

b

correspond exactly to the positive integer components of the binomials.

5

Calculate the final required sum,

a + b

.

2 + 3 = 5

.

The question specifically asks for the sum of the parameters.

Key Concept

Factoring non-monic quadratic trinomials

Question 38Question →

Match each unsimplified algebraic expression with its fully simplified equivalent form. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each unsimplified expression matches exactly one fully simplified form by applying the product, quotient, and power rules of exponents.

The correct matches are found by systematically applying the fundamental properties of exponents: adding exponents for multiplication, subtracting for division, multiplying for powers of powers, and converting negative exponents to reciprocals.

Step-by-Step Solution

1

Simplify the product of powers expression: (3x^-2y^4)(5x^6y^-3)

15x^4y

Multiply numerical coefficients together. Add exponents for variables with the same base according to the product rule.

2

Simplify the quotient of powers expression: (30x^5y^-2) / (2x^9y^3)

15 / (x^4y^5)

Divide numerical coefficients. Subtract the denominator's exponent from the numerator's exponent. Convert negative exponents to positive by moving them to the denominator.

3

Simplify the power of a product expression: (-2x^3y^-1)^4

(16x^12) / y^4

Raise every factor inside the parentheses to the 4th power. Note that an even power of a negative coefficient yields a positive result.

4

Simplify the expression with an outer negative exponent: (3x^-2 / y^4)^-2

(x^4y^8) / 9

Multiply all inner exponents by the outer exponent -2. Then take the reciprocal of terms with negative exponents to rewrite them with positive exponents.

Key Concept

Operations with Variable Exponents

Question 39Question →

Consider the polynomial $8x^2 - 10x - 3$ . When this expression is factored completely into the form $(4x + a)(2x - b)$ , where $a$ and $b$ are positive integers, what are the corresponding values for $a$ and $b$ ?

Fill in the blanks below

The value of

a

is and the value of

b

is .

Show answer & explanation

Answer

The value of a is 1 and the value of b is 3.

By applying the AC method, the quadratic

8x^2 - 10x - 3

can be rewritten as

8x^2 - 12x + 2x - 3

and factored by grouping into

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

. Aligning this with

(4x + a)(2x - b)

directly shows that

a = 1

and

b = 3

.

Step-by-Step Solution

1

Identify the type of polynomial and the appropriate factoring method.

The polynomial

8x^2 - 10x - 3

is a non-monic quadratic trinomial (where the leading coefficient is not 1). We can use the ac-method.

The ac-method involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

2

Calculate the ac product and find the target sum.

The product is

a \times c = 8 \times (-3) = -24

. The target sum is the middle coefficient, which is

-10

.

These values are required to split the middle term for factoring by grouping.

3

Find two integers that multiply to -24 and add to -10.

The numbers

-12

and

2

meet these conditions, since

(-12) \times 2 = -24

and

(-12) + 2 = -10

.

This allows us to rewrite the middle term

-10x

without changing the value of the polynomial.

4

Rewrite the middle term and factor by grouping.

8x^2 - 12x + 2x - 3

. Grouping the terms yields

(8x^2 - 12x) + (2x - 3)

. Factoring out the greatest common factor from each group gives

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

.

Grouping transforms the four-term expression into a format where a common binomial factor can be extracted.

5

Factor out the common binomial factor and determine the final values.

Factoring out

(2x - 3)

leaves

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

. Comparing this to the given form

(4x + a)(2x - b)

reveals that

a = 1

and

b = 3

.

Matching the derived factorization with the required format identifies the specific constants requested by the problem.

Key Concept

Factoring non-monic quadratic trinomials using the AC method and grouping.

Question 40Question →

Find the completely factored form of the given polynomial:

-12x^2 + 30x + 72

Which of the following options shows the correct factorization?

-6(2x - 3)(x + 4)

-6(2x + 3)(x - 4)

-2(2x + 3)(3x - 12)

6(2x + 3)(x - 4)

Show answer & explanation

Answer: $-6(2x + 3)(x - 4)$

Answer

The completely factored form is

-6(2x + 3)(x - 4)

.

The polynomial requires factoring out the Greatest Common Factor (GCF) of -6, which leaves the trinomial

2x^2 - 5x - 12

. Factoring this trinomial yields the binomials

(2x + 3)

and

(x - 4)

. Multiplying these by the GCF gives the fully factored expression.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the polynomial

-12x^2 + 30x + 72

.

The GCF is -6. Factoring this out gives:

-6(2x^2 - 5x - 12)

.

Factoring out a negative GCF makes the leading coefficient of the trinomial positive, which simplifies the subsequent factoring steps.

2

Factor the remaining quadratic trinomial

2x^2 - 5x - 12

by finding two integers that multiply to

(2 \times -12) = -24

and add to the middle coefficient

-5

.

The two integers are -8 and 3.

These integers will be used to split the middle term to factor by grouping.

3

Rewrite the trinomial by splitting the middle term using the integers found, then factor by grouping.

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

.

Grouping allows us to factor out common binomials to find the factored form of the trinomial.

4

Combine the GCF with the factored binomials to write the final completely factored expression.

-6(2x + 3)(x - 4)

A completely factored polynomial must include all extracted common factors alongside the irreducible binomial components.

Key Concept

Factoring polynomials using a Greatest Common Factor (GCF) and factoring non-monic trinomials.

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