Sum and Difference of Cubes Practice Test

Factor and apply the identities for a³+b³ and a³−b³.

Sum and Difference of Cubes Practice Test

This test has 20 questions

Cubic Sign Tapestry

Perfect cubes factor by pattern. The sign pattern decides the quadratic factor.

Sum and difference of cubes problems become much easier when you separate recognition from factoring. First remove any greatest common factor. Then identify the real cube roots of both terms, keep the original sign in the binomial factor, reverse the middle sign in the quadratic factor, and keep the last term positive.

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

Perfect cubesCheck coefficients and variable exponents before choosing an identity.

GCF firstRemove any common factor before looking for a cubic identity.

SOAP signsSame, Opposite, Always Positive keeps the factors organized.

Verify by expansionThe quadratic factor should rebuild exactly the original two cubes.

1. SOAP is a sign map for both cube identities

The binomial factor keeps the original sign. The quadratic factor reverses the middle sign and keeps the final square positive.

Sum of cubesSame sign outside, opposite middle sign inside.

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

Sameplus

Oppositeminus

Always Positivelast term

Difference of cubesSame sign outside, opposite middle sign inside.

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

Sameminus

Oppositeplus

Always Positivelast term

2. Recognition starts with real cube roots

Both numerical coefficients and variable powers must be perfect cubes before the identity applies directly.

Cube-root scannerIdentify the two cube roots before thinking about signs or factors.

Sum structure

8 x^{3} + 27 = {(2 x)}^{3} + 3^{3}

Both terms are exact cubes.

Difference structure

64 x^{3} - 125 y^{3} = {(4 x)}^{3} - {(5 y)}^{3}

Coefficients and variables both cube cleanly.

Negative cube roots are real

\sqrt[3]{216} = 6, \sqrt[3]{- 125} = - 5

Odd roots preserve the sign of a negative real number.

3. Once the cube roots are known, substitute them into the identity

Do not cube or expand again. Build the binomial and quadratic factors directly.

Sum of cubes example

8 x^{3} + 27 = (2 x + 3) (4 x^{2} - 6 x + 9)

The binomial uses addition; the quadratic middle term uses subtraction.

Difference of cubes example

64 x^{3} - 125 y^{3}

(4 x - 5 y) (16 x^{2} + 20 x y + 25 y^{2})

The binomial uses subtraction; the quadratic middle term uses addition.

4. Remove a greatest common factor before applying a cube identity

Factoring the GCF can reveal a clean sum or difference of cubes that was not obvious at first glance.

GCF first

If a common numerical or variable factor is present, remove it before using the cubic identity. Otherwise the factorization is incomplete.

Sum after removing the GCF

12 x^{3} + 96

12 (x^{3} + 8)

12 (x + 2) (x^{2} - 2 x + 4)

Difference after removing the GCF

15 x^{3} - 120

15 (x^{3} - 8)

15 (x - 2) (x^{2} + 2 x + 4)

The common factor stays in front of the final cubic factorization.

5. Coefficients and two variables stay attached to their cube roots

Treat each complete cube root as one quantity before forming the factors.

Two-variable sum

27 a^{3} + 8 b^{3}

(3 a + 2 b) (9 a^{2} - 6 a b + 4 b^{2})

The middle term of the quadratic factor is the product of the two complete cube roots.

Two-variable difference

64 m^{3} - 125 n^{3}

(4 m - 5 n) (16 m^{2} + 20 m n + 25 n^{2})

The last term remains positive even when the original cubes are subtracted.

6. Expansion is the fastest verification that the SOAP signs are correct

The mixed terms should cancel, leaving only the two original cubes.

Check the sum factorization

(a + b) (a^{2} - a b + b^{2})

a^{3} + b^{3}

The mixed terms cancel after multiplication.

verify⇄

Check the difference factorization

(a - b) (a^{2} + a b + b^{2})

a^{3} - b^{3}

The same cancellation confirms the opposite middle sign.

7. Symmetric cube expressions can collapse because matching terms cancel

These identities are useful when two binomial cubes appear with opposite internal signs.

Symmetric sum

{(a + b)}^{3} + {(a - b)}^{3} = 2 a^{3} + 6 a b^{2}

The odd-power terms in the second quantity cancel, leaving a simpler expression.

Symmetric difference

{(a + b)}^{3} - {(a - b)}^{3} = 6 a^{2} b + 2 b^{3}

The complementary terms cancel instead, leaving the remaining mixed structure.

8. Error analysis: most cube-factor mistakes are sign or structure mistakes

SOAP helps, but the identity still requires genuine perfect cubes and complete factorization.

Middle sign not reversed

$a^{3} + b^{3} \neq (a + b) (a^{2} + a b + b^{2})$

Final quadratic term made negative

$a^{3} - b^{3} \neq (a - b) (a^{2} + a b - b^{2})$

GCF left inside

Always remove a common factor before applying the cube identity.

Coefficient not recognized as a perfect cube

Take the real cube root of the numerical coefficient before building the factors.

Quadratic factor treated as another binomial cube

The second factor has three terms and follows the SOAP sign pattern.

Sum of squares mistaken for a cube identity

$a^{2} + b^{2}$ is not a sum-of-cubes pattern.

Final cube-factor audit

Before accepting a factorization, verify the GCF, cube roots, SOAP signs, and expansion check.

Did I remove the greatest common factor first?The cubic identity should be applied only after common factors are extracted.

Are both remaining terms perfect cubes?Check numerical coefficients and variable exponents.

Did the binomial keep the original sign?This is the Same part of SOAP.

Did the quadratic middle term use the opposite sign?This is the Opposite part of SOAP.

Is the final quadratic term positive?This is the Always Positive part of SOAP.

Did I expand backward to verify the two original cubes?The mixed terms should cancel completely.