MK · Lesson test

Algebraic Expressions Lesson Test

This short test checks whether you can simplify and evaluate expressions accurately, not just recognize the rules. It contains 15 original questions in the ASVAB Mathematics Knowledge style, based on the Algebraic Expressions lesson: combining like terms, distributing across parentheses, substituting negative values, expanding binomials and factoring.

15-question lesson test
24 more practice questions
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Lesson test

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Question 1 of 15 · easy

In the expression $7 a^{2} - 4 b + 9$ , what is the coefficient of b?

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Answer: C

Correct answer: −4. The term containing b is $- 4 b$ . The number multiplying b, including its sign, is −4.

A: This drops the minus sign, but the sign in front of a term is part of its coefficient.
B: 9 is the constant term; it has no variable attached.
C (correct): The coefficient is the number multiplying b, and the minus sign in front of the term belongs to it.
D: 7 is the coefficient of $a^{2}$ , not of b.

Question 2 of 15 · easy

Which term is a like term with $3 x y^{2}$ ?

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Answer: A

Correct answer: -8xy². Like terms need identical variables with identical exponents. Only $- 8 x y^{2}$ has x to the first power and y to the second power; the coefficients do not matter.

A (correct): It has exactly the same variable part, x to the first power and y squared; only the coefficient differs.
B: Both letters are squared here, while x is to the first power in $3 x y^{2}$ .
C: The y in this term is only to the first power, so the powers do not match.
D: The coefficient matches, but here x is squared instead of y, so the variable parts differ.

Question 3 of 15 · easy

Which expression is equivalent to $8 p - 3 + 2 p + 11$ ?

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Answer: A

Correct answer: 10p + 8. Group like terms:

8 p + 2 p = 10 p and - 3 + 11 = 8

The result is $10 p + 8$ .

A (correct): Adding the p terms gives 10p and adding the constants −3 and 11 gives 8.
B: This treats −3 as +3; the sign in front of the 3 belongs to that term.
C: This adds the constants to the p terms, but a constant is not a like term with p.
D: This subtracts 2p from 8p, but the 2p term is being added.

Question 4 of 15 · easy

Which expression is equivalent to $- 4 (3 - 2 y)$ ?

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Answer: B

Correct answer: -12 + 8y. Distribute −4 to each term:

- 4 \cdot 3 = - 12 and - 4 \cdot (- 2 y) = 8 y

The result is $- 12 + 8 y$ .

A: This gets the signs of both products backward.
B (correct): Multiplying −4 by 3 gives −12 and −4 by −2y gives +8y.
C: This multiplies only the first term by −4 and leaves −2y unchanged.
D: This forgets that a negative times a negative is positive in the second product.

Question 5 of 15 · easy

What is the value of $5 n - 8$ when

n = -3

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Answer: B

Correct answer: −23. Put the value in parentheses:

5 (- 3) - 8 = - 15 - 8 = - 23

A: This uses 5 times 3 as +15, dropping the negative sign of n.
B (correct): Substituting gives $5 (- 3) - 8 = - 15 - 8 = - 23$ .
C: This has the right size but loses the negative sign of the result.
D: This adds 8 to −15 instead of subtracting 8.

Question 6 of 15 · medium

What is the simplified form of $3 (2 x - 5) - 2 (x - 4)$ ?

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Answer: C

Correct answer: 4x - 7. Distribute each factor:

3 (2 x - 5) = 6 x - 15 and - 2 (x - 4) = - 2 x + 8

Combine:

6 x - 2 x = 4 x and - 15 + 8 = - 7

so the result is $4 x - 7$ .

A: This multiplies −2 by −4 to get −8 instead of +8.
B: This multiplies −2 only into x and leaves −4 unchanged inside.
C (correct): Distributing gives $6 x - 15 - 2 x + 8$ , which combines to $4 x - 7$ .
D: This adds 2x instead of subtracting it after distributing −2.

Question 7 of 15 · medium

x = -3

what does the polynomial $2 x^{2} + 5 x - 3$ equal?

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Answer: C

Correct answer: 0. Substitute with parentheses: $2 (- 3)^{2} + 5 (- 3) - 3$ .

Square first:

(- 3)^{2} = 9 so 2 (9) = 18 then 18 - 15 - 3 = 0

A: This treats $(- 3)^{2}$ as −9, but a squared negative number is positive.
B: This multiplies 2 by −3 first and then squares, which ignores the order of operations.
C (correct): Squaring first gives $2 (9) + 5 (- 3) - 3 = 18 - 15 - 3 = 0$ .
D: This treats $5 (- 3)$ as +15 instead of −15.

Question 8 of 15 · medium

What is $\frac{3}{4} m - \frac{1}{6} m$ in simplest form?

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Answer: B

Correct answer: 7/12·m. Rewrite over 12:

\frac{3}{4} = \frac{9}{12} and \frac{1}{6} = \frac{2}{12} then \frac{9}{12} - \frac{2}{12} = \frac{7}{12}

so the result is $\frac{7}{12} m$ .

A: This subtracts the numerators and adds the denominators, which is not how fractions combine.
B (correct): With a common denominator of 12, $\frac{9}{12} - \frac{2}{12} = \frac{7}{12}$ .
C: This adds the fractions instead of subtracting them.
D: This subtracts 3 − 1 over 12 without first rewriting each fraction with the common denominator.

Question 9 of 15 · medium

What is the product $(- 3 x^{2} y^{3}) (4 x y^{2})$ ?

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Answer: A

Correct answer: -12x³y⁵. Multiply coefficients:

- 3 \times 4 = - 12

Add exponents for each letter:

x^{2 + 1} = x^{3} and y^{3 + 2} = y^{5}

The product is $- 12 x^{3} y^{5}$ .

A (correct): The coefficients give −12, x has exponents 2 + 1 = 3, and y has exponents 3 + 2 = 5.
B: The exponents are right, but a negative times a positive coefficient is negative.
C: This adds the coefficients −3 and 4 instead of multiplying them.
D: This multiplies the exponents instead of adding them.

Question 10 of 15 · medium

Which expression equals $(3 x + 2) (x - 4)$ when expanded?

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Answer: B

Correct answer: 3x² - 10x - 8. Multiply each term by each term:

3 x \cdot x = 3 x^{2} 3 x \cdot (- 4) = - 12 x 2 \cdot x = 2 x and 2 \cdot (- 4) = - 8

Combine the middle terms:

- 12 x + 2 x = - 10 x

The result is $3 x^{2} - 10 x - 8$ .

A: This keeps only the first and last products and drops the outer and inner terms.
B (correct): The four products are $3 x^{2}$ , −12x, 2x and −8, and the middle terms combine to −10x.
C: The middle term has the wrong sign; −12x + 2x is −10x.
D: This uses −2x for the inner product, but 2 times x is +2x.

Question 11 of 15 · medium

Which expression shows $18 a^{4} - 12 a^{2}$ with its greatest common factor factored out?

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Answer: D

Correct answer: 6a²(3a² - 2). The largest number dividing 18 and 12 is 6, and the highest power of a in both terms is $a^{2}$ , so the GCF is $6 a^{2}$ .

Divide each term:

18 a^{4} \div 6 a^{2} = 3 a^{2} and 12 a^{2} \div 6 a^{2} = 2

The result is $6 a^{2} (3 a^{2} - 2)$ .

A: This takes the highest power of a instead of the lowest power common to both terms.
B: The sign inside is wrong; distributing back would give $+ 12 a^{2}$ .
C: Dividing $12 a^{2}$ by $6 a^{2}$ leaves 2, not 2a; distributing back would give $12 a^{3}$ .
D (correct): The GCF is $6 a^{2}$ , and dividing each term by it leaves $3 a^{2}$ and −2.

Question 12 of 15 · hard

What is the simplified form of $\frac{x ^{2} - 25}{3 x - 15}$ , and which value of x must be excluded?

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Answer: D

Correct answer: (x + 5)/3, with x ≠ 5. Factor both parts:

x^{2} - 25 = (x - 5) (x + 5) and 3 x - 15 = 3 (x - 5)

Cancel the common factor $x - 5$ to get $\frac{x + 5}{3}$ . The original denominator is zero when x = 5, so x = 5 is excluded.

A: This cancels the wrong factor; the denominator contains x − 5, so x + 5 is what remains on top.
B: The simplified form is right, but x = −5 does not make $3 x - 15$ zero; x = 5 does.
C: This forgets the factor 3 that comes out of the denominator $3 x - 15$ .
D (correct): Factoring gives $\frac{( x - 5 ) ( x + 5 )}{3 ( x - 5 )}$ , and the original denominator is zero at x = 5.

Question 13 of 15 · hard

Which expression is equivalent to $(x + 3)^{2} - (x - 3)^{2}$ ?

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Answer: C

Correct answer: 12x. Use the identities:

(x + 3)^{2} = x^{2} + 6 x + 9 and (x - 3)^{2} = x^{2} - 6 x + 9

Subtract, flipping every sign of the second:

x^{2} + 6 x + 9 - x^{2} + 6 x - 9 = 12 x

A: This squares each binomial as $x^{2} + 9$ , forgetting the middle term 2ab.
B: This adds the second square instead of subtracting it.
C (correct): Expanding gives $(x^{2} + 6 x + 9) - (x^{2} - 6 x + 9)$ , and everything cancels except 12x.
D: This flips only the first sign after the minus, so the 6x terms cancel and the 9s add.

Question 14 of 15 · hard

What is the complete factorization of $2 y^{3} - 50 y$ ?

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Answer: A

Correct answer: 2y(y - 5)(y + 5). The GCF of both terms is 2y:

2 y^{3} - 50 y = 2 y (y^{2} - 25)

Since

y^{2} - 25 = y^{2} - 5^{2}

it factors as $(y - 5) (y + 5)$ . The complete factorization is $2 y (y - 5) (y + 5)$ .

A (correct): Pulling out 2y leaves $y^{2} - 25$ , a difference of squares that factors as $(y - 5) (y + 5)$ .
B: A difference of squares factors as a sum times a difference, not as a squared binomial.
C: This uses 25 itself instead of its square root 5 in each factor.
D: This loses the common factor y; multiplying back gives $2 y^{2} - 50$ , not the original.

Question 15 of 15 · hard

For

x \neq = 0

which expression is equivalent to $\frac{15 x ^{3} - 10 x ^{2} + 5 x}{5 x} - (x - 4)$ ?

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Answer: D

Correct answer: 3x² - 3x + 5. Divide each term by 5x: $3 x^{2} - 2 x + 1$ .

Then subtract the binomial, flipping both signs:

3 x^{2} - 2 x + 1 - x + 4 = 3 x^{2} - 3 x + 5

A: This flips only the first sign after the minus and subtracts 4 instead of adding it.
B: This adds x instead of subtracting it when removing the parentheses.
C: This treats 5x divided by 5x as 0 instead of 1.
D (correct): The quotient is $3 x^{2} - 2 x + 1$ ; subtracting x − 4 gives $- 2 x - x = - 3 x$ and $1 + 4 = 5$ .

More practice

24 extra questions

Practice 1 of 24 · easy

How many terms are in the expression $4 x^{3} - 2 x + 7 y - 1$ ?

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Answer: D

Correct answer: 4. Terms are the pieces joined by plus or minus signs. Here they are $4 x^{3}$ , $- 2 x$ , $7 y$ and $- 1$ , so there are 4 terms. A constant counts as a term.

A: This counts only the terms with variables and skips the constant -1.
B: This counts something extra, such as the exponent, as a separate term.
C: This counts only the different letters, x and y, instead of the terms.
D (correct): Plus and minus signs split it into four terms: 4x^3, -2x, 7y and -1.

Practice 2 of 24 · easy

A clerk totals two shipments with the expression $6 k + 3 - 4 k + 9$ . After combining like terms, what is the simplified result?

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Answer: B

Correct answer: 2k + 12. Group like terms:

6 k - 4 k = 2 k and 3 + 9 = 12

The result is $2 k + 12$ . Terms with k and constant terms cannot be merged.

A: This combines the k terms with the constants, which are not like terms.
B (correct): The k terms give 6k - 4k = 2k and the constants give 3 + 9 = 12.
C: This adds 4k instead of subtracting it; the minus sign belongs to 4k.
D: This subtracts 9 from 3, but the 9 is being added.

Practice 3 of 24 · easy

A supply room has 7 bins, each holding $(2 t - 3)$ parts. Which expression gives the total, $7 (2 t - 3)$ , without parentheses?

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Answer: B

Correct answer: 14t - 21. The distributive property says the outside factor multiplies every term inside:

7 \cdot 2 t = 14 t and 7 \cdot (- 3) = - 21 so 7 (2 t - 3) = 14 t - 21

A: This multiplies 7 by the first term only and leaves -3 unchanged.
B (correct): Multiply 7 by both terms: 7 x 2t = 14t and 7 x (-3) = -21.
C: This adds 7 to each number instead of multiplying.
D: This multiplies correctly but loses the minus sign on the second term.

Practice 4 of 24 · medium

Remove the parentheses and collect terms in $12 - (5 r - 8)$ . What do you get?

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Answer: D

Correct answer: 20 - 5r. A minus sign before parentheses acts like multiplying by -1, so every sign inside flips: $12 - 5 r + 8$ . Combine the constants:

12 + 8 = 20

The result is $20 - 5 r$ .

A: This flips only the first sign and keeps -8, giving 12 - 5r - 8.
B: This combines 12 and -5r as if they were like terms.
C: This handles the 8 correctly but leaves +5r, so the r term was never flipped.
D (correct): The minus sign flips both inside terms: 12 - 5r + 8 = 20 - 5r.

Practice 5 of 24 · easy

A thermostat formula gives a reading of $6 - 2 m$ . If

m = -4

what reading does the formula give?

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Answer: C

Correct answer: 14. Substitute in parentheses: $6 - 2 (- 4)$ . Multiply first:

- 2 \times - 4 = 8 then 6 + 8 = 14

A: This drops the sign of -4 and computes 6 - 8.
B: This subtracts first, 6 - 2 = 4, and then multiplies by -4, breaking the order of operations.
C (correct): 6 - 2(-4) = 6 + 8 = 14.
D: This computes only -2(-4) and forgets to include the 6.

Practice 6 of 24 · hard

Working from the innermost grouping outward, simplify $7 - 3 [2 y - (y - 4)]$ . What is the result?

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Answer: A

Correct answer: -3y - 5. Work from the inside out. Inner part:

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

Distribute -3: $7 - 3 y - 12$ . Combine constants: $- 3 y - 5$ .

A (correct): Inside: 2y - y + 4 = y + 4. Then 7 - 3y - 12 = -3y - 5.
B: This forgets to flip -4 in the inner parentheses, getting y - 4 and then 7 - 3y + 12.
C: This subtracts 7 - 3 first, but the 3 must multiply the bracket before any subtraction.
D: This makes a sign slip at the end: 7 - 12 is -5, not 5.

Practice 7 of 24 · medium

A shop models its weekly profit, in dollars, as $6 (y + 2) - 4 (2 y - 3)$ . Written in simplest form, what is this profit expression?

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Answer: D

Correct answer: 24 - 2y. Distribute each factor:

6 (y + 2) = 6 y + 12 and - 4 (2 y - 3) = - 8 y + 12

Combine:

6 y - 8 y = - 2 y and 12 + 12 = 24

giving $24 - 2 y$ .

A: This gives the last product as -12 instead of +12; -4 times -3 is positive.
B: This adds 8y instead of subtracting it, ignoring the minus in front of the 4.
C: This multiplies -4 by 2y but only subtracts 3 from 12, leaving out -4 times -3.
D (correct): 6y + 12 - 8y + 12 combines to -2y + 24.

Practice 8 of 24 · medium

What is the value of $x^{2} - 3 x + 4$ when

x = -5

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Answer: C

Correct answer: 44. Substitute with parentheses: $(- 5)^{2} - 3 (- 5) + 4$ . Square first:

(- 5)^{2} = 25 then - 3 (- 5) = 15

Total:

25 + 15 + 4 = 44

A: This squares -5 as -25 instead of 25, giving -25 + 15 + 4.
B: This squares correctly but treats -3(-5) as -15, giving 25 - 15 + 4.
C (correct): (-5)^2 = 25, -3(-5) = 15, and 25 + 15 + 4 = 44.
D: This makes both sign mistakes, giving -25 - 15 + 4.

Practice 9 of 24 · medium

Which expression is equal to $\frac{1}{2} w + \frac{2}{5} w$ ?

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Answer: D

Correct answer: 9/10·w. Rewrite both coefficients over the common denominator 10:

\frac{1}{2} = \frac{5}{10} and \frac{2}{5} = \frac{4}{10}

Add the numerators:

\frac{5}{10} + \frac{4}{10} = \frac{9}{10}

so the result is $\frac{9}{10} w$ .

A: This adds the tops and the bottoms separately, which is not how fractions add.
B: This multiplies the fractions instead of adding them.
C: This changes 1/2 to 5/10 but keeps 2 as the top of the second fraction instead of 4.
D (correct): With tenths, 5/10 + 4/10 = 9/10.

Practice 10 of 24 · hard

Simplify $\frac{x ^{2} - 49}{x ^{2} + 7 x}$ . Which answer gives the simplified form and the values of x that must be excluded?

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Answer: B

Correct answer: (x - 7)/x, with x ≠ 0 and x ≠ -7. Factor both parts:

x^{2} - 49 = (x - 7) (x + 7) and x^{2} + 7 x = x (x + 7)

Cancel the common factor $x + 7$ to get $\frac{x - 7}{x}$ . The original denominator is zero when x = 0 or

x = -7

so both values are excluded.

A: The form is right, but -7 also makes the original denominator zero, so it must be excluded too.
B (correct): Factoring gives (x - 7)(x + 7) over x(x + 7); cancel x + 7, and both 0 and -7 make the original bottom zero.
C: This cancels the x^2 terms and divides -49 by 7x, canceling terms instead of factors.
D: This cancels the wrong factor; the x + 7 factor is the one shared by top and bottom.

Practice 11 of 24 · hard

Expand both products and collect terms. What does $(x + 4)^{2} - (x + 4) (x - 4)$ simplify to?

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Answer: A

Correct answer: 8x + 32. Use the identities:

(x + 4)^{2} = x^{2} + 8 x + 16 and (x + 4) (x - 4) = x^{2} - 16

Subtract, flipping every sign of the second:

x^{2} + 8 x + 16 - x^{2} + 16 = 8 x + 32

A (correct): x^2 + 8x + 16 minus (x^2 - 16) leaves 8x + 32.
B: This subtracts x^2 - 16 but fails to flip the sign of -16, so the constants cancel.
C: This writes (x + 4)^2 as x^2 + 16, forgetting the middle term 8x.
D: This adds the second product instead of subtracting it.

Practice 12 of 24 · medium

Which expression equals $(x - 7) (x + 2)$ when multiplied out?

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Answer: A

Correct answer: x² - 5x - 14. Multiply every term by every term: first $x^{2}$ , outer $2 x$ , inner $- 7 x$ , last $- 14$ . Combine the middle terms:

2 x - 7 x = - 5 x

The result is $x^{2} - 5 x - 14$ .

A (correct): Products are x^2, 2x, -7x and -14; the middle terms combine to -5x.
B: This multiplies only the first terms and the last terms, skipping the outer and inner products.
C: This flips the sign of the middle term; 2x - 7x is -5x.
D: This subtracts 2x instead of adding it when combining the middle terms.

Practice 13 of 24 · medium

Which shows $12 m^{4} - 20 m^{3} + 8 m^{2}$ with its greatest common factor factored out?

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Answer: C

Correct answer: 4m²(3m² - 5m + 2). The largest number dividing 12, 20 and 8 is 4, and the lowest power of m in any term is $m^{2}$ , so the GCF is $4 m^{2}$ . Divide each term: $3 m^{2}$ , $- 5 m$ and $2$ . Check by distributing back.

A: This is equivalent but not fully factored; 2m is a common factor, not the greatest one.
B: This drops the last term; 8m^2 divided by 4m^2 is 2, not 0.
C (correct): 4 is the largest number dividing 12, 20 and 8, and m^2 is the highest power of m in every term.
D: Distributing back gives 8m, not 8m^2, so the last term is wrong.

Practice 14 of 24 · medium

Which is the factored form of $4 a^{2} - 81$ ?

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Answer: A

Correct answer: (2a - 9)(2a + 9). Both parts are perfect squares:

4 a^{2} = (2 a)^{2} and 81 = 9^{2}

A difference of squares factors as a difference times a sum: $(2 a - 9) (2 a + 9)$ . Multiplying back, the middle terms cancel.

A (correct): 4a^2 is (2a)^2 and 81 is 9^2, so the difference of squares gives (2a - 9)(2a + 9).
B: This squares a difference, which gives 4a^2 - 36a + 81, adding a middle term.
C: This squares a sum, which gives 4a^2 + 36a + 81, and the constant would be positive.
D: This splits 4a^2 into 4a times a, which gives a middle term of 27a.

Practice 15 of 24 · medium

For y not equal to 0, which expression is equivalent to $\frac{18 y ^{4} - 6 y ^{3} + 3 y ^{2}}{3 y ^{2}}$ ?

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Answer: B

Correct answer: 6y² - 2y + 1. Divide each term separately by $3 y^{2}$ :

\frac{18 y ^{4}}{3 y ^{2}} = 6 y^{2} \frac{- 6 y ^{3}}{3 y ^{2}} = - 2 y and \frac{3 y ^{2}}{3 y ^{2}} = 1

The result is $6 y^{2} - 2 y + 1$ .

A: This treats 3y^2 divided by 3y^2 as 0, but anything nonzero divided by itself is 1.
B (correct): Dividing each term by 3y^2 gives 6y^2, -2y and 1.
C: This divides the last term by y^2 only and forgets to divide the 3 by 3.
D: This adds the exponents, which is the rule for multiplying, not dividing.

Practice 16 of 24 · medium

For x not equal to 2 or -2, which expression is equivalent to $\frac{5 x + 10}{x ^{2} - 4}$ ?

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Answer: C

Correct answer: 5/(x - 2). Factor the top with the GCF:

5 x + 10 = 5 (x + 2)

Factor the bottom as a difference of squares:

x^{2} - 4 = (x - 2) (x + 2)

Cancel the common factor $x + 2$ to get $\frac{5}{x - 2}$ .

A: This cancels the x - 2 factor, but the shared factor is x + 2.
B: This cancels a 2 out of single terms across plus and minus signs, which is not allowed.
C (correct): Top is 5(x + 2) and bottom is (x - 2)(x + 2); canceling x + 2 leaves 5 over x - 2.
D: This places x - 2 in the top instead of the bottom after canceling.

Practice 17 of 24 · medium

A square garden has a side of $(2 x + 5)$ feet. Which expression gives its area, $(2 x + 5)^{2}$ , in square feet?

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Answer: B

Correct answer: 4x² + 20x + 25. Use

(a + b)^{2} = a^{2} + 2 ab + b^{2}

with a = 2x and b = 5:

(2 x)^{2} = 4 x^{2} 2 (2 x) (5) = 20 x and 5^{2} = 25

The area is $4 x^{2} + 20 x + 25$ .

A: This squares each term but leaves out the middle term 2ab.
B (correct): Using (a + b)^2 = a^2 + 2ab + b^2: 4x^2, 2(2x)(5) = 20x, and 25.
C: This uses ab for the middle term instead of 2ab.
D: This forgets to square the coefficient 2 in the first term.

Practice 18 of 24 · easy

What is the constant term in $9 x^{2} - x - 12$ ?

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Answer: B

Correct answer: -12. A constant is a term with no variable. In $9 x^{2} - x - 12$ , the only such term is $- 12$ , and the minus sign in front of it is part of the term.

A: This drops the sign; the sign in front of a term belongs to that term.
B (correct): The constant is the term with no variable, and its minus sign belongs to it.
C: This is the coefficient of x^2, not a constant term.
D: This is the coefficient of x, the number hidden in front of -x.

Practice 19 of 24 · medium

After distributing and combining like terms, what is the simplest form of $3 (x - 2) - (x - 6)$ ?

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Answer: A

Correct answer: 2x. Distribute:

3 (x - 2) = 3 x - 6

and the minus before the second group flips both signs: $- x + 6$ . Combine:

3 x - x = 2 x and - 6 + 6 = 0

so the result is $2 x$ . Checking with x = 1: the original is

- 3 + 5 = 2

which matches.

A (correct): 3x - 6 - x + 6 gives 2x, and the constants cancel.
B: This flips only the first sign in the second group, writing -x - 6 instead of -x + 6.
C: This multiplies 3 by x only, writing 3x - 2 instead of 3x - 6.
D: This drops the minus sign in front of the second group and adds x - 6.

Practice 20 of 24 · hard

What is the value of $2 a^{2} - ab + b^{2}$ when

a = -3

and b = 2?

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Answer: C

Correct answer: 28. Substitute with parentheses: $2 (- 3)^{2} - (- 3) (2) + (2)^{2}$ . Square first:

2 (9) = 18

Next,

(- 3) (2) = - 6

and subtracting it adds 6. Last,

2^{2} = 4

Total:

18 + 6 + 4 = 28

A: This squares -3 as -9, giving -18 + 6 + 4.
B: This treats -ab as -6 instead of +6, giving 18 - 6 + 4.
C (correct): 2(9) - (-6) + 4 = 18 + 6 + 4 = 28.
D: This doubles before squaring, using (2a)^2 = 36 instead of 2a^2 = 18.

Practice 21 of 24 · easy

Which expression is equal to $(3 x^{2}) (5 x^{4})$ ?

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Answer: A

Correct answer: 15x⁶. To multiply monomials, multiply the coefficients and add the exponents of the same variable:

3 \times 5 = 15 and x^{2 + 4} = x^{6}

The product is $15 x^{6}$ .

A (correct): Multiply the coefficients, 3 x 5 = 15, and add the exponents, 2 + 4 = 6.
B: This multiplies the exponents instead of adding them.
C: This adds the coefficients instead of multiplying them.
D: This adds the coefficients and multiplies the exponents, reversing both rules.

Practice 22 of 24 · hard

Factor the numerator and the denominator of $\frac{3 x ^{2} - 12}{x ^{2} - 2 x}$ , then cancel common factors, with x not equal to 0 or 2. What remains?

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Answer: C

Correct answer: (3(x + 2))/x. Factor the top completely:

3 x^{2} - 12 = 3 (x^{2} - 4) = 3 (x - 2) (x + 2)

Factor the bottom:

x^{2} - 2 x = x (x - 2)

Cancel the common factor $x - 2$ to get $\frac{3 ( x + 2 )}{x}$ .

A: This cancels x + 2 as if it were in the bottom; the shared factor is x - 2.
B: This loses the common factor 3 that was pulled out of the top.
C (correct): Top factors to 3(x - 2)(x + 2), bottom to x(x - 2); cancel x - 2.
D: This also cancels the x in the bottom, but x is not a factor of the top.

Practice 23 of 24 · hard

A storage bin's volume is the product of $(2 x - 3)$ and $(x^{2} + 4 x - 1)$ . Which polynomial in standard form gives this product?

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Answer: D

Correct answer: 2x³ + 5x² - 14x + 3. Multiply each term of the first factor by every term of the second. $2 x$ gives $2 x^{3} + 8 x^{2} - 2 x$ . $- 3$ gives $- 3 x^{2} - 12 x + 3$ . Combine like terms:

8 x^{2} - 3 x^{2} = 5 x^{2} and - 2 x - 12 x = - 14 x

so the result is $2 x^{3} + 5 x^{2} - 14 x + 3$ .

A: This adds 3x^2 instead of subtracting it; -3 times x^2 is -3x^2.
B: This leaves out the -12x that comes from -3 times 4x.
C: This gets the last sign wrong; -3 times -1 is +3.
D (correct): 2x gives 2x^3 + 8x^2 - 2x; -3 gives -3x^2 - 12x + 3; combining gives 2x^3 + 5x^2 - 14x + 3.

Practice 24 of 24 · medium

For which value of x is the expression $\frac{x + 5}{2 x - 8}$ undefined?

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Answer: D

Correct answer: x = 4. A fraction is undefined when its denominator is zero. Set

2 x - 8 = 0 so 2 x = 8 and x = 4

A zero in the numerator only makes the value 0.

A: This makes the top zero, which gives a value of 0, not an undefined expression.
B: This sets x equal to the constant 8 without dividing by the coefficient 2.
C: This makes a sign slip; 2(-4) - 8 is -16, not 0.
D (correct): The bottom 2x - 8 is zero when x = 4, and division by zero is undefined.

About this test

An explanation accompanies each answer. It shows the simplification line by line and points out which distractor results from a common slip, such as dropping a negative sign during distribution or combining terms with different powers.

When you see your result, look for patterns rather than just the total. Several misses on substitution suggest rereading the sections on evaluating with powers and signed values. Factoring errors point to the GCF and difference-of-squares sections. Twenty-four additional practice questions are available for this lesson if you want more work before linear equations.

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Algebraic Expressions Lesson Test

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