Verify whether the polynomial identity $(x + y + z)^2 - (x + y - z)^2 = 4z(x + y)$ holds for all real polynomials, and use it to expand and simplify fully the expression $((2x^2 - 3x) + 4)^2 - ((2x^2 - 3x)

Verify whether the polynomial identity $(x + y + z)^2 - (x + y - z)^2 = 4z(x + y)$ holds for all real polynomials, and use it to expand and simplify fully the expression $((2x^2 - 3x) + 4)^2 - ((2x^2 - 3x) - 4)^2$ .

Final answer

$32x^2 - 48x$

Steps

Verify the polynomial identity

To verify that $(x + y + z)^2 - (x + y - z)^2 = 4z(x + y)$ , group the terms by setting $A = x + y$ and $B = z$ .

The left-hand side of the identity becomes: $(A + B)^2 - (A - B)^2$

Recall the algebraic expansions for the square of a sum and the square of a difference: $(A + B)^2 = A^2 + 2AB + B^2$ $(A - B)^2 = A^2 - 2AB + B^2$

Subtracting the second expression from the first: $(A + B)^2 - (A - B)^2 = (A^2 + 2AB + B^2) - (A^2 - 2AB + B^2)$ $(A + B)^2 - (A - B)^2 = A^2 + 2AB + B^2 - A^2 + 2AB - B^2 = 4AB$

Alternatively, factoring as a difference of two squares gives: $[(A + B) - (A - B)][(A + B) + (A - B)] = (2B)(2A) = 4AB$

Substituting $A = x + y$ and $B = z$ back into $4AB$ : $4AB = 4(x + y)z = 4z(x + y)$

Since this simplification is valid for any algebraic expressions, the identity holds for all real polynomials.

Apply the identity to the given expression

We are asked to simplify: $((2x^2 - 3x) + 4)^2 - ((2x^2 - 3x) - 4)^2$

Match this directly to the general form $(A + B)^2 - (A - B)^2 = 4AB$ , where:

$A = 2x^2 - 3x$
$B = 4$

Applying the verified formula: $((2x^2 - 3x) + 4)^2 - ((2x^2 - 3x) - 4)^2 = 4 \cdot A \cdot B$ $= 4(2x^2 - 3x)(4)$ $= 16(2x^2 - 3x)$

Expand and simplify fully

Distribute $16$ across the binomial: $16(2x^2 - 3x) = 16 \cdot 2x^2 - 16 \cdot 3x = 32x^2 - 48x$

Common Pitfall: Expanding each binomial fully before subtracting leads to lengthy calculations involving fourth-degree polynomials where sign errors are frequent. Recognizing the structure $(A+B)^2 - (A-B)^2 = 4AB$ allows immediate reduction of high-degree terms.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student misremembers the identity factor as 2 instead of 4 from confusing (A + B)² - (A - B)² with (A + B)² + (A - B)², resulting in 16x² - 24x.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
(x+y+z)^2 - (x+y-z)^2 = ((x+y)+z)^2 - ((x+y)-z)^2 $\implies$ (x+y)^2 + 2z(x+y) + z^2 - ((x+y)^2 - 2z(x+y) + z^2) = 4z(x+y) $\implies$ $\text{The identity holds.}$
Step 2
$\text{Let }$ x+y = 2x^2-3x $\text{ and }$ z=4 $\implies$ $\text{Expression}$ = 2z(x+y) = 2(4)(2x^2-3x)
Correct
$\text{Let }$ x+y = 2x^2-3x $\text{ and }$ z=4 $\implies$ $\text{Expression}$ = 4z(x+y) = 4(4)(2x^2-3x)
You accurately matched the components of the expression to the identity, but you wrote the identity expression as $2z(x+y)$ instead of the verified $4z(x+y)$ .
Step 3
= 8(2x^2-3x) = 16x^2 - 24x
Correct
= 16(2x^2-3x) = 32x^2 - 48x
Your algebraic expansion and distribution were computed accurately based on your expression from the previous step, but the final answer is halved due to the earlier missing factor of 2.

WHAT YOU DID WELL

Compliments from Your Teacher

You verified the algebraic identity accurately using binomial expansion.

You correctly identified the corresponding sub-expressions for $(x + y)$ and $z$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Formula retrieval error

You proved that the identity yields $4z(x + y)$ , but then switched the coefficient from $4$ to $2$ when setting up the substitution.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the coefficient of the cross-term in a single binomial expansion ( $2AB$ ) with the result of subtracting two squared binomials, $(A+B)^2 - (A-B)^2 = 4AB$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Check the cross-term difference: $(A + B)^2 - (A - B)^2 = (A^2 + 2AB + B^2) - (A^2 - 2AB + B^2) = 4AB$
Always recall that subtracting leaves $+2AB - (-2AB) = 4AB$ , giving a coefficient of 4, not 2.

CORRECTED PATH

ACEpath Teacher's Demonstration

Substitute $x+y = 2x^2 - 3x$ and $z = 4$ into the verified formula $4z(x + y)$ :
$\text{Expression} = 4(4)(2x^2 - 3x) = 16(2x^2 - 3x)$
Distribute $16$ across the binomial terms: $16 \cdot 2x^2 - 16 \cdot 3x = 32x^2 - 48x$ .

Final answer

$32x^2 - 48x$

NEXT STEPS

Final Takeaways from Your Teacher

Double-check your copied formula against your verified proof before substituting values.
Practice distinguishing between $(A+B)^2 - (A-B)^2 = 4AB$ and $(A+B)^2 + (A-B)^2 = 2(A^2 + B^2)$ .

OverviewACEpath Teacher's Diagnosis

02 The student forgets to distribute the multiplication to both terms inside the parentheses, producing 32x² - 3x.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
[(x + y + z) - (x + y - z)][(x + y + z) + (x + y - z)] = (2z)(2(x + y)) = 4z(x + y) $\implies$ $\text{The identity holds.}$
Step 2
$\text{Let }$ x + y = 2x^2 - 3x $\text{ and }$ z = 4 $\implies$ 4(4)(2x^2 - 3x) = 16(2x^2 - 3x)
Step 3
= 32x^2 - 3x
Correct
16(2x^2 - 3x) = 32x^2 - 48x
When multiplying $16$ by $(2x^2 - 3x)$ , you multiplied $16 \times 2x^2 = 32x^2$ , but you did not multiply $16 \times (-3x)$ , leaving it as $-3x$ instead of $-48x$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You used the difference of two squares to verify the identity cleanly and efficiently.

You correctly matched the terms of the algebraic identity to simplify the target expression into $16(2x^2 - 3x)$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete distribution across parentheses

You distributed $16$ to the first term $2x^2$ to get $32x^2$ , but missed multiplying the second term $-3x$ by $16$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of the multiplier across multi-term expressions inside parentheses during algebraic expansion.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distributive Law Check: For any real numbers or expressions $a$ , $b$ , and $c$ , $a(b - c) = a \cdot b - a \cdot c$ .
Draw distribution arrows or write out each intermediate product: $16 \cdot (2x^2) - 16 \cdot (3x)$ before combining.

CORRECTED PATH

ACEpath Teacher's Demonstration

Apply the distributive property fully across both terms: $16 \cdot 2x^2 - 16 \cdot 3x$ .
Compute the numerical products: $16 \cdot 2 = 32$ and $16 \cdot 3 = 48$ .

Final answer

32x^2 - 48x

NEXT STEPS

Final Takeaways from Your Teacher

Double-check that every term inside parentheses is multiplied by the outer factor before declaring the expansion complete.
Practice distributing two-digit constants across binomials to build a consistent habit of checking both terms.

OverviewACEpath Teacher's Diagnosis

03 The student mistakes the identity coefficient 4z for z², computing 4²(2x² - 3x) without the factor of 4 and getting 16(2x² - 3x) leading to 16x² - 48x via an arithmetic error.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
((x + y) + z)^2 - ((x + y) - z)^2 = 4z(x + y) $\implies$ $\text{The identity holds.}$
Correct
[(x + y) + z]^2 - [(x + y) - z]^2 = ((x + y)^2 + 2z(x + y) + z^2) - ((x + y)^2 - 2z(x + y) + z^2) = 4z(x + y)
You asserted that the identity holds after regrouping the terms, but you did not show the required algebraic expansion or difference-of-squares steps to verify it.
Step 2
z = 4 $\text{and}$ x + y = 2x^2 - 3x
Step 3
$\text{Expression} = 4(2x^2 - 3x)$
Correct
$\text{Expression}$ = 4z(x + y) = 4(4)(2x^2 - 3x) = 16(2x^2 - 3x)
You substituted $z = 4$ as just a single factor of $4$ , omitting the leading constant $4$ from $4z(x + y)$ .
Step 4
= 8x^2 - 12x
Correct
= 32x^2 - 48x
Your expansion distributed your factor of $4$ correctly, but the result was incorrect due to the missing factor from the previous step.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly regrouped the terms to show the structure of the identity.

You accurately identified that $z = 4$ and $x + y = 2x^2 - 3x$ for the substitution.

You correctly expanded your binomial by distributing the leading factor.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted Coefficient during Variable Substitution

You wrote $4(2x^2 - 3x)$ instead of $4(4)(2x^2 - 3x)$ , replacing the entire product $4z$ with just $4$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

When substituting a numerical value into a monomial with a numerical coefficient (such as $4z$ with $z = 4$ ), you conflated the existing coefficient with the substituted value instead of evaluating their product $4 \cdot z$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Substitute with Parentheses: Write each variable inside parentheses before simplifying: $4z \to 4(z)$ .
Replace $z$ with $(4)$ so the product $4(4) = 16$ is explicitly visible before distributing.

CORRECTED PATH

ACEpath Teacher's Demonstration

$\text{Expression}$ = 4z(x + y) = 4(4)(2x^2 - 3x)
= 16(2x^2 - 3x)
= 32x^2 - 48x

Final answer

32x^2 - 48x

NEXT STEPS

Final Takeaways from Your Teacher

Write variable substitutions explicitly in parentheses, writing $4z$ as $4 \cdot (4)$ rather than just $4$ .
Always provide explicit expansion or factoring steps when a problem asks you to verify an identity.

Lock it in

Question

What does the expression $(A + B)^2 - (A - B)^2$ simplify to?

Answer

$4AB$ . The squared terms $A^2$ and $B^2$ cancel, leaving $2AB - (-2AB) = 4AB$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student misremembers the identity factor as 2 instead of 4 from confusing (A + B)² - (A - B)² with (A + B)² + (A - B)², resulting in 16x² - 24x.

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 forgets to distribute the multiplication to both terms inside the parentheses, producing 32x² - 3x.

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 mistakes the identity coefficient 4z for z², computing 4²(2x² - 3x) without the factor of 4 and getting 16(2x² - 3x) leading to 16x² - 48x via an arithmetic error.

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

Similar question