📝 Multiply Integers in algebraic expressions (21 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 21 questions available

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What is Multiply Integers in algebraic expressions?

Definition:
Multiplying integers in algebraic expressions means finding the product of signed integer coefficients and constants, where the result's sign depends on the signs of the factors: positive product for same signs and negative product for opposite signs, applied when simplifying or expanding expressions.

Working:
Multiply the absolute values of the integers, then assign the sign: if both factors have same sign (both positive or both negative), product is positive; if signs differ, product is negative; then combine with any variables present, multiplying coefficients as usual.

Example:
Multiply $(-5)$ and $2$ in expression $(-5)(2x)$ .
Solution: $(-5) \times 2 = -10$ (different signs, negative), so $(-5)(2x) = -10x$ .

Reason:
This operation is critical in polynomial multiplication, factoring, and solving equations, as it allows us to distribute terms and combine like terms correctly, which is a building block for more advanced algebra topics.

Easy

Medium

Hard

📝 All Multiply Integers in algebraic expressions MCQs

Q1. A student evaluates $(-7)(-4)$ by saying, "Both numbers are negative, so the product must be negative." What is the correct value and reasoning?

-28

, because two negative signs remain negative

28

, because multiplying two integers with the same sign gives a positive product ✅

-11

, because the signs are combined before multiplying

11

, because the negative signs cancel the larger magnitude

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When two integers have the same sign, their product is positive. The absolute values are multiplied first: $7\times4=28$ . Therefore, $(-7)(-4)=28$ . The student's mistake comes from incorrectly assuming every expression containing negative numbers produces a negative result.

Q2. Which expression has the greatest value?

(-8)(3)

(-6)(-5)

✅

7(-4)

(-9)(2)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Evaluate each product carefully: $(-8)(3)=-24$ , $(-6)(-5)=30$ , $7(-4)=-28$ , and $(-9)(2)=-18$ . The only positive product is $30$ , so option B is greatest. This requires comparing both sign and magnitude rather than multiplying mechanically.

Q3. A submarine changes its position by $-6$ meters during each of 5 identical time intervals. If the same change continues, which expression represents the total change and what does its result mean?

5(-6)=30

, so it rises 30 meters

5(-6)=-30

, so it moves 30 meters downward ✅

(-5)(-6)=-30

, so it moves downward

(-6)-5=-11

, so it moves 11 meters downward

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Five identical changes of $-6$ meters can be modeled as $5(-6)$ . Multiplication gives $-30$ , meaning the submarine's position decreases by 30 meters overall. The negative sign represents direction, while the magnitude 30 represents the total distance of the change.

Q4. A student claims that $(-12)(-3)=-36$ because "multiplying a negative number always makes the result negative." Which response best identifies the error?

A.The student should add

12+3

instead

B.The student should ignore both negative signs and use

12\times3=36

, then keep the original negative sign

C.The student incorrectly assumes one negative factor is present; two negative factors produce a positive product ✅

D.The student should subtract

3

from

12

and make the answer negative

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The error is in applying the sign rule incorrectly. There are two negative factors, not one. A product of two integers with the same sign is positive, while their absolute values are multiplied. Thus $(-12)(-3)=36$ , not $-36$ .

Q5. A number-line model shows five equal jumps, each of length 4 units. The arrows point left for a positive multiplier and right for a negative multiplier. Which product could the model represent if the final position is 20 units to the right of the starting point?

5(4)=20

5(-4)=-20

(-5)(4)=-20

(-5)(-4)=20

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The model ends 20 units to the right, so its value is $+20$ . A negative multiplier reverses the direction of the repeated movement. Thus five negative groups of $-4$ produce $(-5)(-4)=20$ . The result demonstrates why two negative factors create a positive product.

Q6. A temperature changes by $-3^\circ\text{C}$ each hour for 4 hours, and then the entire change is reversed. Which expression and result correctly describe the reversed total change?

4(-3)=-12

, then

-(-12)=12

✅

4(-3)=12

, then

-(12)=-12

(-4)(-3)=-12

, then

-(-12)=-12

4(3)=12

, then

12

remains unchanged

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: First calculate the original change: $4(-3)=-12^\circ\text{C}$ . Reversing that change means taking its opposite, so $-(-12)=12^\circ\text{C}$ . The final result is positive because the reversal changes the direction of the entire negative change.

Q7. For integers $a$ and $b$ , suppose $a<0$ , $b<0$ , and $|a|>|b|$ . Which statement must always be true about $ab$ ?

ab<0

and

|ab|<|a|

ab>0

, and its magnitude equals

|a||b|

✅

ab=0

because both factors are negative

ab<0

, and its magnitude equals

|a|-|b|

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Since both $a$ and $b$ are negative, their product must be positive. The magnitude of a product equals the product of the magnitudes, so $|ab|=|a||b|$ . The condition $|a|>|b|$ does not change the sign rule or the multiplication process.

Q8. Without calculating the exact magnitude, what can be determined about $(-13)(-8)$ ?

A.The product must be negative because both factors are negative

B.The product must be positive because the factors have the same sign ✅

C.The product must be zero because the signs cancel

D.The product could be positive or negative depending on which factor is larger

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Both factors are negative, so they have the same sign. A product of two integers with the same sign is positive. The magnitudes determine the size of the answer, but they do not change its positive sign.

Q9. A student evaluates $(-9)(6)(-2)$ from left to right. What sign should the final answer have, and why?

A.Negative, because the first two factors have different signs

B.Positive, because there are two negative factors overall ✅

C.Negative, because multiplication always preserves the sign of the first factor

D.Positive, because

6

is larger than both negative factors

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: First, $(-9)(6)$ is negative because the signs differ. Multiplying that negative result by $-2$ gives a positive product because the two remaining factors have different signs. Equivalently, two negative factors overall produce a positive result.

Q10. A factory records a loss of $\$250$ per defective unit. If an accounting correction reverses the sign of this loss for 7 units, which expression represents the corrected total?

7(-250)=-1750

(-7)(250)=-1750

(-7)(-250)=1750

✅

7(250)=-1750

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The loss is represented by $-250$ per unit, while reversing the sign can be represented by multiplying by $-7$ . Thus $(-7)(-250)=1750$ . The positive result represents a net gain or correction of $\$1750$ , requiring interpretation of both signs.

Q11. A student says $(-5)(-12)=-60$ because "two negative signs should make the answer more negative." What is the most precise correction?

A.Two negative factors produce a positive product, so the result is

60

✅

B.Only the larger negative factor determines the sign, so the result is

-60

C.The signs should be added first, giving

0

, so the result is

0

D.Negative numbers can never produce positive products

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The student's error is treating multiplication signs like addition. For multiplication, two factors with the same sign produce a positive product. Therefore, $(-5)(-12)=60$ . The magnitudes $5$ and $12$ are multiplied after determining the positive sign.

Q12. A graph shows four points representing products: $A=(-4)(3)$ , $B=(-4)(-3)$ , $C=(4)(-3)$ , and $D=(4)(3)$ . Which point lies on the positive side of the vertical axis if the vertical coordinate represents the product?

A.Only

A

B.Only

C

B

and

D

✅

A

and

C

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The products are $A=-12$ , $B=12$ , $C=-12$ , and $D=12$ . Therefore, $B$ and $D$ have positive vertical coordinates. The graph interpretation depends on recognizing that same-sign factors produce positive products while different-sign factors produce negative products.

Q13. Two students use different methods for $(-6)(-4)(5)$ . Student 1 groups the negative factors first, while Student 2 multiplies the first two factors and then the third. What should they conclude?

A.Student 1 is correct because only grouping negative factors is valid

B.Student 2 is correct because negative factors cannot be grouped

C.Both methods must give

120

because the two negative factors create a positive product ✅

D.The methods must give different signs because the multiplication order changes

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Student 1 can calculate $(-6)(-4)=24$ , then $24(5)=120$ . Student 2 reaches the same result by multiplying sequentially. Multiplication allows regrouping without changing the product, and the two negative factors create a positive intermediate result.

Q14. For nonzero integers $a$ , $b$ , and $c$ , suppose $a<0$ , $b>0$ , and $c<0$ . Without knowing their magnitudes, what must be true about $abc$ ?

A.The product must be negative because one factor is positive

B.The product must be positive because there are two negative factors ✅

C.The product must be zero because the signs cancel

D.The sign cannot be determined without knowing the magnitudes

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: There are two negative factors, $a$ and $c$ , and one positive factor, $b$ . Multiplying the two negative factors produces a positive value, and multiplying that positive result by $b>0$ remains positive. Magnitudes affect size, not the sign of a nonzero product.

Q15. What is the result of multiplying $-17$ by $-1$ , and what does the operation do to the original number?

A.It gives

-17

because multiplying by

-1

keeps the sign unchanged

B.It gives

17

because multiplying by

-1

reverses the sign ✅

C.It gives

0

because the two negative signs cancel completely

D.It gives

-18

because

1

is subtracted from the number

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying any integer by $-1$ changes its sign while preserving its magnitude. Therefore, $(-17)(-1)=17$ . The operation does not add or subtract $1$ ; instead, it produces the additive opposite of the original number.

Q16. A student claims that $x(-1)=x$ because "multiplying by 1 does not change a number." What is the key error?

A.The student should divide by

-1

instead

B.The negative sign was ignored; multiplying by

-1

produces the opposite of

x

✅

C.The result should always be

0

D.The value of

x

must be positive for multiplication to work

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student has ignored the negative sign in $-1$ . While multiplying by $1$ leaves a number unchanged, multiplying by $-1$ reverses its sign. Thus $x(-1)=-x$ , regardless of whether $x$ is positive, negative, or zero.

Q17. A bank account adjustment is recorded as $-\$450$ . A correction reverses the direction of the adjustment by multiplying it by $-1$ . What is the corrected value?

-\$450

\$449

\$450

✅

D.0

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The original adjustment is $-450$ . Reversing its direction means multiplying by $-1$ : $(-450)(-1)=450$ . The positive result represents a credit or increase of $\$450$ , so the correction changes both the sign and interpretation of the amount.

Q18. If $a=-8$ and $b=5$ , which expression correctly compares $a(-1)$ and $b(-1)$ ?

A.Both products are negative, so

a(-1)<b(-1)

B.Both products are positive, so

a(-1)=b(-1)

C.The products are

8

and

-5

, so

a(-1)>b(-1)

✅

D.The products are

-8

and

5

, so

a(-1)<b(-1)

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Multiplying by $-1$ reverses each sign. Thus $a(-1)=8$ and $b(-1)=-5$ . Since $8>-5$ , the transformed value of $a$ is greater than the transformed value of $b$ , even though $a<b$ originally.

Q19. A number-line graph represents $x=6$ . A transformation sends every point $x$ to $x(-1)$ . Where should the point move?

A.From

6

-6

, reflecting its position across zero ✅

B.From

6

7

, moving one unit right

C.From

6

0

, because the sign disappears

D.From

6

3

, because the magnitude is halved

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Multiplication by $-1$ changes $6$ into $-6$ . On a number line, this places the point the same distance from zero but on the opposite side. Thus the transformation reverses direction while preserving distance from the origin.

Q20. An expression is simplified as $(-1)(-3x+7)$ . Which result is correct, and why?

-3x+7

, because multiplication by

-1

changes nothing

3x-7

, because every term changes sign ✅

-3x-7

, because only the constant changes

3x+7

, because the negative sign disappears only from the variable term

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiplying the entire expression by $-1$ changes the sign of every term. Therefore, $(-1)(-3x+7)=3x-7$ . A common error is changing the sign of only one term instead of applying the multiplier to the complete expression.

Q21. For a nonzero integer $n$ , suppose multiplying $n$ by $-1$ produces a value that is greater than $n$ . What must be true about $n$ ?

n

must be positive

n

must be negative ✅

n

must equal zero

n

could be any nonzero integer

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The condition is $-n>n$ . Subtracting $n$ gives $-2n>0$ , so $n<0$ . Therefore, $n$ must be negative. For example, if $n=-4$ , then multiplying by $-1$ gives $4$ , which is greater than $-4$ .}

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