📝 Simplify Expressions Using the Distributive Property (15 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 15 questions available

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What is Simplify Expressions Using the Distributive Property?

Definition:
Simplifying expressions using the distributive property involves multiplying a single term by each term inside parentheses, expressed as $a(b+c) = ab + ac$ , which removes grouping symbols and allows combining like terms, making expressions simpler and easier to evaluate.

Working:
Multiply the term outside the parentheses by every term inside, paying attention to signs; then combine any like terms that result from the distribution, simplifying the expression to its standard form without parentheses.

Example:
Simplify $-2(3x - 4) + 5$ .
Solution: Distribute: $-2 \times 3x + (-2) \times (-4) = -6x + 8$ ; then add 5: $-6x + 13$ .

Reason:
The distributive property is a key tool for expanding, factoring, and simplifying algebraic expressions, and it is essential for solving linear equations, working with polynomials, and understanding function transformations.

Easy

Medium

Hard

📝 All Simplify Expressions Using the Distributive Property MCQs

Q1. A student simplifies $4(2x-3)+2x$ as $8x-3+2x=10x-3$ . Which expression correctly identifies the student's mistake and gives the simplified result?

A.The student multiplied only the first term; the result is

10x-12

B.The student forgot to distribute

4

-3

; the result is

10x-12

C.The student should distribute

4

to both terms; the result is

10x-12

. ✅

D.The student should combine

2x

with

-3

; the result is

10x-3

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The factor $4$ must multiply every term inside the parentheses, including $-3$ . Thus $4(2x-3)=8x-12$ , and adding $2x$ gives $10x-12$ . The distractors reflect common errors involving incomplete distribution or combining unlike terms.

Q2. Which expression is equivalent to $5(2x+7)-3x$ after distributing and combining like terms?

7x+35

✅

10x+32

10x+35-3x

7x+32

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First distribute $5$ to both terms inside the parentheses, giving $10x+35$ . Then combine the like terms $10x$ and $-3x$ , producing $7x+35$ . This requires applying distribution correctly before simplifying the resulting expression.

Q3. A student claims that $4(x+6)=4x+6$ because the $4$ only needs to multiply the variable. Which explanation best evaluates the claim?

A.The claim is correct because constants do not need distribution.

B.The claim is incorrect because

4

must multiply both

x

and

6

, giving

4x+24

. ✅

C.The claim is incorrect because the result should be

x+24

D.The claim is correct only when

x

is positive.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The factor outside parentheses multiplies every term inside them. Therefore $4(x+6)=4x+24$ , not $4x+6$ . The error comes from distributing to only one term, a common misconception that changes the value of the entire expression.

Q4. A shop charges $x$ dollars for one item and adds a fixed service fee of $5$ dollars to every purchase. A customer buys $6$ items. Which expression correctly models the total cost and its expanded form?

6(x+5)=6x+30

✅

6x+5=6(x+5)

x+30=6(x+5)

6(x-5)=6x-30

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Each of the six items is associated with the item cost $x$ and the service amount $5$ , so the total can be modeled as $6(x+5)$ . Distributing $6$ gives $6x+30$ , preserving the meaning of the situation.

Q5. Two students simplify $7(3x-2)$ . Student A writes $21x-14$ , while Student B writes $21x-2$ . Who is correct, and why?

A.Student A, because

7

must multiply both terms. ✅

B.Student B, because constants should not be multiplied.

C.Both students are correct for different values of

x

D.Neither is correct because the expression cannot be simplified.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Student A correctly distributes $7$ to both $3x$ and $-2$ . Thus $7(3x-2)=21x-14$ . Student B distributes the factor only to the variable term, leaving the constant unchanged, which violates the required multiplication across every term.

Q6. A graph represents $y=3(x+2)$ . A student says its expanded form is $y=3x+2$ . Which feature of the graph would reveal the student's error?

A.The graph would have

y

-intercept

2

instead of

6

. ✅

B.The graph would have slope

2

instead of

3

C.The graph would pass through the origin in both forms.

D.The graph would have slope

6

instead of

3

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Distributing gives $y=3x+6$ , so the correct graph has slope $3$ and $y$ -intercept $6$ . The student's expression $3x+2$ keeps the slope but changes the intercept to $2$ , allowing the error to be detected graphically.

Q7. Which pair of methods gives the same simplified result for $8(2x+3)-4(x+5)$ ?

A.Method 1:

16x+24-4x+20

; Method 2:

12x+44

B.Method 1:

16x+24-4x-20

; Method 2:

12x+4

✅

C.Method 1:

16x+3-4x-5

; Method 2:

12x-2

D.Method 1:

16x+24+4x+20

; Method 2:

20x+44

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Distributing carefully gives $8(2x+3)=16x+24$ and $-4(x+5)=-4x-20$ . Combining like terms produces $12x+4$ . The negative factor must distribute to both terms, making this a useful test of sign accuracy and method comparison.

Q8. For an unknown number $x$ , a student notices that $9(x+4)$ and $9x+36$ always have the same value. Which reasoning best explains why this equivalence must hold for every value of $x$ ?

A.The

9

changes only the value of

x

, so the constant remains

4

B.The

9

can be distributed to each addend, producing

9x+36

, so both expressions represent the same calculation. ✅

C.The expressions are equal only when

x=4

D.The expressions are equal because

9+4=13

for every

x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The factor $9$ multiplies every addend inside the parentheses. Therefore $9(x+4)=9x+36$ for every value of $x$ . This is not dependent on a particular value of $x$ ; it follows from applying multiplication to each part of the sum.

Q9. Which expression is equivalent to $6(2x-5)+3x$ after removing the parentheses and combining like terms?

15x-30

✅

12x-30+3x

9x-5

15x-5

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Distribute $6$ to both terms inside the parentheses: $6(2x-5)=12x-30$ . Then combine $12x$ and $3x$ to obtain $15x-30$ . The other choices reflect incomplete distribution or incorrect combination of unlike terms.

Q10. A student simplifies $-4(x-7)$ as $-4x-28$ . What is the best analysis of the student's reasoning?

A.The student is correct because both terms become negative.

B.The student should distribute

-4

to both terms, giving

-4x+28

. ✅

C.The student should remove the negative sign before distributing.

D.The correct result is

4x-28

because opposites cancel.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The factor $-4$ must multiply both terms. Thus $(-4)(x)=-4x$ and $(-4)(-7)=+28$ , so the correct result is $-4x+28$ . The sign changes because multiplying two negative numbers produces a positive result.

Q11. A rectangular playground has length $4(x+3)$ meters and width $2x$ meters. Which expression represents its perimeter after removing parentheses and simplifying?

6x+12

12x+24

✅

8x+12

12x+12

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The perimeter is $2[L+W]$ , so $P=2[4(x+3)+2x]$ . Distributing gives $4x+12+2x=6x+12$ , and multiplying by $2$ gives $12x+24$ . This requires modeling the situation before simplifying.

Q12. The expression $5-(2x-9)$ is rewritten by a student as $5-2x-9$ . Which correction gives the equivalent simplified expression?

-2x-4

2x+14

-2x+14

✅

2x-4

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Subtracting the entire quantity $2x-9$ is equivalent to multiplying it by $-1$ . Therefore $5-(2x-9)=5-2x+9=-2x+14$ . The student's error is failing to reverse the sign of the $-9$ term.

Q13. A line has equation $y=-3(x-2)$ . Which expanded equation describes the same line, and what does its graph reveal about the constant term?

y=-3x-6

, with

y

-intercept

-6

y=-3x+6

, with

y

-intercept

6

✅

y=3x-6

, with

y

-intercept

-6

y=3x+6

, with

y

-intercept

6

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Distributing $-3$ gives $-3x+6$ . Therefore the graph has slope $-3$ and crosses the $y$ -axis at $6$ . A sign error in the constant would move the graph to a different vertical position even though the slope remains unchanged.

Q14. Which pair of expressions represents opposites of each other after simplification?

3(x-4)

and

-3x+12

2(x+5)

and

-2x+10

4(x-2)

and

-4x+8

5(x+1)

and

-5x-5

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Simplifying $5(x+1)$ gives $5x+5$ , whose opposite is $-(5x+5)=-5x-5$ . The other pairs are also related through sign changes or distribution, but they do not correctly form an opposite pair as written.

Q15. For which value of $x$ does $3(x-4)$ have the same value as the opposite of $2x+6$ ?

x=-18

x=6

x=-2

✅

x=18

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Set $3(x-4)=-(2x+6)$ . Distributing gives $3x-12=-2x-6$ . Adding $2x$ and $12$ to both sides gives $5x=6$ , so $x=\frac{6}{5}$ , which is not listed. Therefore the options are inconsistent, and a careful solver should reject all four choices rather than force an incorrect answer.

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