A common student misconception states that 'multiplication always distributes over every algebraic operation.' (a) Evaluate the difference $D_1(x) = 3(x \cdot y) - (3x \cdot 3y)$ for $x = 2$ and $y = 5$. (b) Expand and simplify the true distributive operation $D_2(x) = -5x(2x - 7) - (3 - x)(4x + 1)$, demonstrating how the negative sign distributes across both terms of the product.

A common student misconception states that 'multiplication always distributes over every algebraic operation.' (a) Evaluate the difference $D_1(x) = 3(x \cdot y) - (3x \cdot 3y)$ for $x = 2$ and $y = 5$ . (b) Expand and simplify the true distributive operation $D_2(x) = -5x(2x - 7) - (3 - x)(4x + 1)$ , demonstrating how the negative sign distributes across both terms of the product.

Final answer

(a) $D_1(2) = 30 - 90 = -60$ ; (b) $D_2(x) = -6x^2 + 24x - 3$

Steps

Part (a): Testing the Misconception on Multiplication

The distributive property states that multiplication distributes over addition and subtraction: $a(b + c) = ab + ac \quad \text{and} \quad a(b - c) = ab - ac$ It does not distribute over multiplication. That is, $a(b \cdot c) \neq (ab) \cdot (ac)$ , because multiplication is associative: $a(bc) = (ab)c = abc$ .

Substitute $x = 2$ and $y = 5$ into the correct product: $3(x \cdot y) = 3(2 \cdot 5) = 3(10) = 30$
Substitute the values into the incorrectly distributed expression: $(3x \cdot 3y) = (3 \cdot 2) \cdot (3 \cdot 5) = 6 \cdot 15 = 90$
Calculate the difference: $D_1(2) = 30 - 90 = -60$

Because the difference is non-zero, this confirms that multiplication does not distribute over multiplication; doing so introduces an extra factor of $3$ .

Part (b): Expanding and Simplifying Using the Distributive Property

Given: $D_2(x) = -5x(2x - 7) - (3 - x)(4x + 1)$

Distribute $-5x$ across the first binomial $(2x - 7)$ : $-5x(2x - 7) = (-5x)(2x) + (-5x)(-7) = -10x^2 + 35x$ (Pitfall: Remember that multiplying two negative terms yields a positive product: $(-5x) \cdot (-7) = +35x$ .)
Expand the second product $(3 - x)(4x + 1)$ : $(3 - x)(4x + 1) = 3(4x + 1) - x(4x + 1)$ $= (12x + 3) - (4x^2 + x)$ $= -4x^2 + 11x + 3$
Distribute the negative sign across the expanded second product: $-(3 - x)(4x + 1) = -(-4x^2 + 11x + 3) = 4x^2 - 11x - 3$
Combine like terms: $D_2(x) = (-10x^2 + 35x) + (4x^2 - 11x - 3)$ $D_2(x) = (-10x^2 + 4x^2) + (35x - 11x) - 3$ $D_2(x) = -6x^2 + 24x - 3$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student distributes the leading negative sign only to the first term of the second expanded product rather than across all terms.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
3(2 $\cdot$ 5) - (3 $\cdot$ 2 $\cdot$ 3 $\cdot$ 5) = 3(10) - (6 $\cdot$ 15) $\implies$ 30 - 90 = -60
Step 2
-5x(2x - 7) = -10x^2 + 35x
Step 3
(3 - x)(4x + 1) = 12x + 3 - 4x^2 - x = -4x^2 + 11x + 3
Step 4
-5x(2x - 7) - (3 - x)(4x + 1) = -10x^2 + 35x - (-4x^2) + 11x + 3 $\implies$ -10x^2 + 35x + 4x^2 + 11x + 3
Correct
-10x^2 + 35x - (-4x^2 + 11x + 3) = -10x^2 + 35x + 4x^2 - 11x - 3
The minus sign precedes the entire expanded expression $(-4x^2 + 11x + 3)$ . You applied the subtraction only to the leading term $-4x^2$ , leaving $+11x$ and $+3$ unchanged instead of negating all three terms to get $+4x^2 - 11x - 3$ .
Step 5
= -6x^2 + 46x + 3
Correct
= -6x^2 + 24x - 3
This final expression collected terms correctly based on your previous line, but it was affected by the missed sign distribution.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly substituted values and carried out the arithmetic to demonstrate that multiplication does not distribute over multiplication.

You distributed the monomial $-5x$ across $(2x - 7)$ with proper sign handling.

You expanded the binomial product $(3 - x)(4x + 1)$ accurately into standard form.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Partial Distributive Subtraction

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating a leading negative sign before a compound multi-term expression as attached only to the first term rather than distributing the factor $-1$ across every term inside parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Keep Parentheses When Subtracting Products: $-(A + B + C) = -A - B - C$
Always place the entire expanded product inside parentheses after a minus sign before distributing $-1$ .
Flip Every Sign in the Group: Multiplying by $-1$ flips the sign of every single term inside the group.
Check each term individually after dropping parentheses to confirm every sign was inverted.

CORRECTED PATH

ACEpath Teacher's Demonstration

Rewrite the subtraction using grouping brackets: $D_2(x) = (-10x^2 + 35x) - (-4x^2 + 11x + 3)$
Distribute the negative sign to every term inside the second group: $-(-4x^2) - (11x) - (3) = +4x^2 - 11x - 3$
Combine like terms: $(-10x^2 + 4x^2) + (35x - 11x) - 3 = -6x^2 + 24x - 3$

Final answer

-6x^2 + 24x - 3

NEXT STEPS

Final Takeaways from Your Teacher

Retain protective parentheses around any expanded polynomial that is preceded by a minus sign.
Practice distributing $-1$ across three-term polynomials before combining like terms.

OverviewACEpath Teacher's Diagnosis

02 The student misapplies the distributive property to a product by thinking 3(x · y) equals (3x · 3y), resulting in an erroneous difference of 0.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
3(x $\cdot$ y) = 3x $\cdot$ 3y $\implies$ D_1(x) = (3 $\cdot$ 2 $\cdot$ 3 $\cdot$ 5) - (3 $\cdot$ 2 $\cdot$ 3 $\cdot$ 5) = 90 - 90 = 0
Correct
3(x $\cdot$ y) = 3(2 $\cdot$ 5) = 30 $\text{and}$ (3x $\cdot$ 3y) = 6 $\cdot$ 15 = 90 $\implies$ D_1(2) = 30 - 90 = -60
Multiplication is associative, not distributive over itself: $a(b \cdot c) = (ab) \cdot c = abc$ , not $(ab) \cdot (ac)$ . Substituting $x = 2$ and $y = 5$ gives $3(2 \cdot 5) = 3(10) = 30$ , while $(3 \cdot 2)(3 \cdot 5) = 6 \cdot 15 = 90$ , giving a difference of $30 - 90 = -60$ .
Step 2
-5x(2x - 7) = -10x^2 + 35x $\implies$ (3 - x)(4x + 1) = -4x^2 + 11x + 3
Step 3
-5x(2x - 7) - (3 - x)(4x + 1) = -10x^2 + 35x - (-4x^2 + 11x + 3) $\implies$ -10x^2 + 35x + 4x^2 - 11x - 3 = -6x^2 + 24x - 3

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded $-5x(2x - 7)$ correctly, carefully handling the double negative to obtain $+35x$ .

You expanded the product $(3 - x)(4x + 1)$ without algebraic or sign errors.

You correctly subtracted the polynomial by distributing the negative sign to every term inside the parentheses.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributing Multiplication Over Multiplication

none

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the expression inside the parentheses as if it were a sum rather than a product, misapplying the distributive property $a(b + c) = ab + ac$ to $a(bc)$ , which erroneously duplicates the factor $a$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Operation inside parentheses: Multiplication distributes across addition and subtraction: $a(b \pm c) = ab \pm ac$ .
Check the operation inside parentheses. If it is multiplication, multiply the factors associatively: $a(bc) = abc$ .
Numerical substitution check: When verifying an algebraic identity, compute each side independently before equating them.
Evaluate $3(x \cdot y)$ first using order of operations, then separately calculate $(3x \cdot 3y)$ and subtract.

CORRECTED PATH

ACEpath Teacher's Demonstration

Substitute $x = 2$ and $y = 5$ into $3(x \cdot y)$ : $3(2 \cdot 5) = 3(10) = 30$ .
Substitute $x = 2$ and $y = 5$ into $(3x \cdot 3y)$ : $(3 \cdot 2) \cdot (3 \cdot 5) = 6 \cdot 15 = 90$ .
Subtract the two quantities: $D_1(2) = 30 - 90 = -60$ .

Final answer

(a) $D_1(2) = -60$ ; (b) $D_2(x) = -6x^2 + 24x - 3$

NEXT STEPS

Final Takeaways from Your Teacher

Review the distinction between the associative property of multiplication $a(bc) = (ab)c$ and the distributive property $a(b + c) = ab + ac$ .
Practice evaluating expressions containing nested products by following standard order of operations before attempting algebraic transformations.

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly multiplies signs when distributing -5x over (2x - 7), writing -35x instead of +35x.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
3(2 $\cdot$ 5) - (3 $\cdot$ 2 $\cdot$ 3 $\cdot$ 5) = 3(10) - (6 $\cdot$ 15) $\implies$ = 30 - 90 $\implies$ = -60
Step 2
-5x(2x - 7) = -10x^2 - 35x
Correct
-5x(2x - 7) = -10x^2 + 35x
When distributing $-5x$ to the second term $-7$ , multiplying two negative values yields a positive product: $(-5x)(-7) = +35x$ .
Step 3
(3 - x)(4x + 1) = -4x^2 + 11x + 3
Step 4
-5x(2x - 7) - (3 - x)(4x + 1) = -10x^2 - 35x - (-4x^2 + 11x + 3) $\implies$ = -10x^2 - 35x + 4x^2 - 11x - 3 $\implies$ = -6x^2 - 46x - 3
Correct
(-10x^2 + 35x) - (-4x^2 + 11x + 3) = -6x^2 + 24x - 3
Your algebra following the sign error was executed logically: you distributed the minus sign across the second polynomial correctly, but the earlier $-35x$ led to an incorrect linear coefficient of $-46x$ instead of $+24x$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly evaluated the numerical expression in part (a) to disprove the misconception.

You accurately expanded the binomial product $(3 - x)(4x + 1)$ .

You correctly distributed the negative sign across all three terms of $(-4x^2 + 11x + 3)$ .

You combined your like terms accurately based on your earlier work.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign multiplication error in distribution

Multiplying $(-5x)$ by $(-7)$ resulted in $-35x$ instead of $+35x$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Dropping the double negative when distributing a monomial with a negative coefficient across a subtraction inside parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Track Signs Explicitly: $(-a)(-b) = +ab$
Before writing down the product, explicitly check whether both factors share the same sign.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute $-5x$ correctly: $-5x(2x - 7) = -10x^2 + 35x$ .
Substitute both expanded parts: $(-10x^2 + 35x) - (-4x^2 + 11x + 3)$ .
Distribute the negative sign across the second group: $-10x^2 + 35x + 4x^2 - 11x - 3$ .
Combine like terms: $(-10 + 4)x^2 + (35 - 11)x - 3 = -6x^2 + 24x - 3$ .

Final answer

(a) $D_1(2) = -60$ ; (b) $D_2(x) = -6x^2 + 24x - 3$

NEXT STEPS

Final Takeaways from Your Teacher

Always write out $(-5x) \cdot (-7)$ as an intermediate check when working with negative coefficients.
Practice distributing negative monomials over subtraction expressions to build fluency.

Lock it in

Question

What does the expression $-(a - b + c)$ simplify to when parentheses are removed?

Answer

$-a + b - c$ . A negative sign outside parentheses multiplies every enclosed term by $-1$ , flipping every sign.

Resources

Answer and steps

Final answer

Steps

Part (a): Testing the Misconception on Multiplication

Part (b): Expanding and Simplifying Using the Distributive Property

Typical errors

01 The student distributes the leading negative sign only to the first term of the second expanded product rather than across all terms.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student misapplies the distributive property to a product by thinking 3(x · y) equals (3x · 3y), resulting in an erroneous difference of 0.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student incorrectly multiplies signs when distributing -5x over (2x - 7), writing -35x instead of +35x.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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