Difference of Squares Factoring Practice Test

Factoring Polynomial Algebra · Difference of Squares

A difference of squares is a mirror pattern: two exact squares separated by subtraction produce conjugate factors

This practice set focuses on recognizing A²−B², extracting any GCF first, writing conjugate factors, factoring repeatedly, solving zero-product equations, rejecting sums of squares, recovering missing constants, and using the identity in higher-degree, multivariable, real-number, area, and product contexts.

A² − B²conjugate factorsGCF first repeat factoringzero-product equations missing valuesarea/product applications

The identity works only after both terms are exact squares and the operation between them is subtraction.

Conjugate split

Take square roots first — then write one minus factor and one plus factor

Recognize the squares

25x² − 49

√(25x²)=5x
√49=7

The factors use the square roots, not the original squared terms.

MIRROR
THE
SIGN

Write conjugates

25x² − 49
= (5x − 7)(5x + 7)

The two binomials are identical except for the sign between their terms.

Quick identity shelf

Core rules for difference-of-squares questions

Main identity Difference of squares A² − B² = (A − B)(A + B)

Use the square roots A and B inside the conjugate factors.

Recognition test Three conditions two terms + subtraction + both exact squares

If any one of these conditions fails, this identity does not apply directly.

Equation solving Zero-product property (A−B)(A+B)=0

Set both factors equal to zero so both solution branches are included.

Complete factoring Recheck each factor factor → inspect → factor again if possible

Higher-degree expressions may contain more than one difference-of-squares layer.

Pattern recognition gate

Before factoring, confirm the identity really applies

Two terms

16x² − 81

A difference of squares is a binomial pattern, not a three-term pattern.

Subtraction

16x² − 81 ✓
16x² + 81 ✗

A sum of squares does not use the real-number identity (A−B)(A+B).

Exact squares

16x²=(4x)²
81=9²

Each term must have a real square root that fits the intended factoring context.

GCF before the identity

A common factor may hide the square pattern

12x² − 48

= 12(x² − 4)

The inside expression is now visibly x²−2².

Then factor the squares

Continue until the expression is completely factored

12(x² − 4)

= 12(x − 2)(x + 2)

Removing only the GCF would be incomplete if the instruction asks for complete factorization.

Repeated difference-of-squares layers

Quartic and sixth-degree expressions may factor more than once

1 · Start x⁴ − 16

Both terms are exact squares.

2 · First split (x² − 4)(x² + 4)

The conjugate factors are formed from x² and 4.

3 · Reinspect x² − 4

This factor is itself another difference of squares.

4 · Complete (x−2)(x+2)(x²+4)

Over the real numbers, x²+4 does not factor by this identity.

Do not apply the pattern to a sum

A²+B² is not the same real-number factoring pattern as A²−B²

This distinction is one of the explicit common mistakes on the page.

Difference x² − 25 = (x−5)(x+5)

The subtraction sign activates the conjugate identity.

Sum x² + 25

This does not factor as (x−5)(x+5) over the real numbers because that product would give x²−25.

From factors to equation solutions

Difference-of-squares equations usually create two opposite branches

Factor the equation

x² − 49 = 0
(x−7)(x+7)=0

Both factors matter.

SET
BOTH
= 0

Solve both branches

x−7=0 → x=7
x+7=0 → x=−7

Keeping only one root loses half of the solution set.

Recovering missing constants or values

Known conjugate factors can be multiplied backward to recover the original expression

Known factors

(x − 6)(x + 6)

The factors are conjugates.

Use the identity backward

x² − 6²

The middle terms cancel automatically.

Recover the constant

x² − 36

The missing constant is determined by squaring the conjugate term.

Real-number factorization

The identity can use irrational square roots when real factoring is allowed

Recognize real squares

x² − 2

2 = (√2)²

2 is not an integer square, but it is the square of a real number.

OVER
THE REALS

Factor with radicals

x² − 2
= (x−√2)(x+√2)

The allowed number system determines whether such factors are considered complete.

side A

side B

Area interpretation

A difference of square areas naturally produces the identity

If a smaller square of area B² is removed from a larger square of area A², the remaining area is A²−B². Algebraically, that same quantity factors as the product of the conjugate expressions A−B and A+B.

A² − B² = (A−B)(A+B)

Verification by multiplication

The middle terms must cancel

First products

A·A = A²

This recreates the first square.

Middle cancellation

+AB − AB = 0

Opposite middle terms cancel because the factors are conjugates.

Last product

(−B)(+B)=−B²

The result is exactly A²−B².

Common mistakes from the page

Most errors come from misreading the pattern or stopping one step too early

Original terms used instead of square roots

Writing 25x²−49 as (25x²−49)(25x²+49). Use 5x and 7, the square roots of the terms.

The identity is built from A and B, not A² and B² inside the factors.

Two identical factors written

Writing (A−B)(A−B). Use conjugates: (A−B)(A+B).

The opposite signs are what cancel the middle terms.

Applied to a sum of squares

Factoring A²+B² as (A−B)(A+B). That product equals A²−B², not A²+B².

Check the operation sign before using the identity.

Stopped before complete factorization

Stopping after x⁴−16=(x²−4)(x²+4). Factor x²−4 again as (x−2)(x+2).

Reinspect every factor after each step.

GCF forgotten

Trying the identity before removing a common factor. Extract the GCF first, then inspect the remaining expression.

This often reveals the exact square pattern cleanly.

Only one equation root kept

Solving only A−B=0. Solve both A−B=0 and A+B=0.

The zero-product property requires every factor branch.

Final difference-of-squares checklist

Before selecting an answer, verify the identity, conjugates, completeness, and equation branches

GCF checked first Any common numerical or variable factor has been removed before applying the identity.

Exact squares + subtraction The remaining expression is truly A²−B² rather than a sum or nonsquare binomial.

Conjugates written correctly The factors use square roots and opposite signs: (A−B)(A+B).

Factored and solved completely Each factor is rechecked, and both zero-product branches are kept when solving an equation.

Difference of Squares Practice Test