Polynomial Identities

11th Grade,
12th Grade

Reviewed by Editorial Team

The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.

Learn about Our Editorial Process

| By Thames

Thames

Community Contributor

Quizzes Created: 7682 | Total Attempts: 9,547,133

| Questions: 20 | Updated: Dec 17, 2025

Share on Facebook Share on Twitter Share on Whatsapp Share on Pinterest Share on Email Copy to Clipboard Embed on your website

Question 1 / 20

0 %

0/100

Score 0/100

1) Expand (x + 5)²

X² + 25

X² + 5x + 25

X² + 10x + 25

X² + 20x + 5

x² + 10x + 25, because using the identity (a + b)² = a² + 2ab + b² with a = x and b = 5 produces x² from squaring x, adds 2·x·5 = 10x as the middle term created by doubling the product of the terms, and ends with 25 from squaring 5.

Explanation

Submit

About This Quiz

Ready to strengthen your foundation with the most important polynomial identities? In this quiz, you’ll practice recognizing standard patterns, rewriting expressions, and applying identities to simplify equations. You’ll work with perfect squares, factorizations, and classic algebraic relationships, building intuition for how and why these identities matter. Step by step, you’ll... see moredevelop the fluency needed to use them confidently in broader algebra problems.
see less

2) What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) Simplify (x + 7)(x − 7)

X² − 7x + 7

X² + 49

X² − 49

X² + 7x − 7

x² − 49, because this fits the difference-of-squares identity (a + b)(a − b) = a² − b², giving x² − 7² = x² − 49, with no middle term since +7 and −7 cancel each other.

Explanation

x² − 49, because this fits the difference-of-squares identity (a + b)(a − b) = a² − b², giving x² − 7² = x² − 49, with no middle term since +7 and −7 cancel each other.

Submit

3) Which expression matches (2x + 3)²?

4x² + 9

4x² + 6x + 9

4x² + 12x + 9

4x² + 3x + 9

4x² + 12x + 9, because expanding with (a + b)² = a² + 2ab + b² yields (2x)² = 4x², then 2·(2x)·3 = 12x, and finally 3² = 9, giving the full trinomial.

Explanation

4x² + 12x + 9, because expanding with (a + b)² = a² + 2ab + b² yields (2x)² = 4x², then 2·(2x)·3 = 12x, and finally 3² = 9, giving the full trinomial.

Submit

4) Simplify (4x − 5)(4x + 5)

16x² − 25

16x² + 25

20x² − 5

4x² − 25

16x² − 25, because this is a difference-of-squares pattern where (a − b)(a + b) = a² − b², so with a = 4x and b = 5 the expansion becomes (4x)² − 25 = 16x² − 25, and the linear terms cancel due to symmetry.

Explanation

Submit

5) Using (a − b)², expand (x − 6)².

X² − 6x + 6

X² − 12x + 36

X² − 36x + 12

X² + 12x + 36

x² − 12x + 36, because applying the identity (a − b)² = a² − 2ab + b² with a = x and b = 6 produces x², subtracts 12x from doubling x·6, and adds 36 from squaring 6.

Explanation

x² − 12x + 36, because applying the identity (a − b)² = a² − 2ab + b² with a = x and b = 6 produces x², subtracts 12x from doubling x·6, and adds 36 from squaring 6.

Submit

6) Simplify (2x − 3)(2x + 3).

4x² + 9

4x² − 9

4x² − 6x + 9

4x² + 6x + 9

4x² − 9, because (a − b)(a + b) = a² − b² yields (2x)² − 3² = 4x² − 9, again with no linear term since +3 and −3 cancel.

Explanation

4x² − 9, because (a − b)(a + b) = a² − b² yields (2x)² − 3² = 4x² − 9, again with no linear term since +3 and −3 cancel.

Submit

7) Expand (x + 2)(x² − 2x + 4).

X³ + 8

X³ + 2x² − 4x + 8

X³ + 8x² + 4x

X³ − 2x² + 4x + 8

x³ + 8, because this expression exactly matches the sum-of-cubes identity (a + b)(a² − ab + b²) = a³ + b³ with a = x and b = 2, giving x³ + 2³ = x³ + 8.

Explanation

x³ + 8, because this expression exactly matches the sum-of-cubes identity (a + b)(a² − ab + b²) = a³ + b³ with a = x and b = 2, giving x³ + 2³ = x³ + 8.

Submit

8) Which identity can simplify 100² − 99² mentally?

(a + b)²

(a − b)²

A² − b² = (a − b)(a + b)

(a + b)³

a² − b² = (a − b)(a + b), because substituting a = 100 and b = 99 gives (100 − 99)(100 + 99) = 1·199 = 199, exploiting the difference-of-squares shortcut to avoid long multiplication.

Explanation

a² − b² = (a − b)(a + b), because substituting a = 100 and b = 99 gives (100 − 99)(100 + 99) = 1·199 = 199, exploiting the difference-of-squares shortcut to avoid long multiplication.

Submit

9) Expand (x + 2)³ using (a + b)³ = a³ + 3a²b + 3ab² + b³.

X³ + 3x² + 6x + 8

X³ + 6x² + 12x + 8

X³ + 4x² + 4x + 8

X³ + 8x² + 12x + 6

x³ + 6x² + 12x + 8, because applying (a + b)³ = a³ + 3a²b + 3ab² + b³ yields x³, then 3x²·2 = 6x², then 3x·4 = 12x, and finally 2³ = 8, giving the full four-term expansion.

Explanation

x³ + 6x² + 12x + 8, because applying (a + b)³ = a³ + 3a²b + 3ab² + b³ yields x³, then 3x²·2 = 6x², then 3x·4 = 12x, and finally 2³ = 8, giving the full four-term expansion.

Submit

10) Which expression equals (a + b)²?

A² + b²

A² − 2ab + b²

A² − b²

A² + 2ab + b²

a² + 2ab + b², because the binomial square identity requires squaring each term and doubling their product to create the middle term, distinguishing it from the difference-of-squares form.

Explanation

a² + 2ab + b², because the binomial square identity requires squaring each term and doubling their product to create the middle term, distinguishing it from the difference-of-squares form.

Submit

11) A square garden has side (x + 3) meters. Find its area.

X² + 3x

X² + 9

X² + 6x + 9

2x² + 3

x² + 6x + 9, because area = side² = (x + 3)², and expanding gives x² + 2·x·3 + 9, representing how polynomial expansion models geometric area growth.

Explanation

x² + 6x + 9, because area = side² = (x + 3)², and expanding gives x² + 2·x·3 + 9, representing how polynomial expansion models geometric area growth.

Submit

12) Two square patios have sides x + 2 and x − 2. Combined area?

2x² + 4

2x² + 8

2x² − 8

2x² + 8x + 8

2x² + 8, because summing (x + 2)² = x² + 4x + 4 and (x − 2)² = x² − 4x + 4 yields x² + 4x + 4 + x² − 4x + 4 = 2x² + 8, where the linear terms cancel due to symmetry.

Explanation

2x² + 8, because summing (x + 2)² = x² + 4x + 4 and (x − 2)² = x² − 4x + 4 yields x² + 4x + 4 + x² − 4x + 4 = 2x² + 8, where the linear terms cancel due to symmetry.

Submit

13) The volume of a cube increases when each edge is lengthened by 3. Original edge = x. New volume = ?

X³ + 3x² + 3x + 3

X³ + 9x² + 27x + 27

X³ + 6x² + 9x + 3

(x³ + 3)³

x³ + 9x² + 27x + 27, because the new edge is x + 3 and the volume is (x + 3)³, which expands to x³ + 3x²·3 + 3x·9 + 27 = x³ + 9x² + 27x + 27, illustrating how binomial cubes model volume change.

Explanation

Submit

14) Which identity helps calculate 105² easily?

(a + b)(a − b) = a² − b²

(a − b)² = a² − 2ab + b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)² = a² + 2ab + b²

(a + b)² = a² + 2ab + b², because setting a = 100 and b = 5 gives 100² + 2·100·5 + 25 = 10000 + 1000 + 25 = 11025, showing an efficient mental-math use of the binomial square formula.

Explanation

(a + b)² = a² + 2ab + b², because setting a = 100 and b = 5 gives 100² + 2·100·5 + 25 = 10000 + 1000 + 25 = 11025, showing an efficient mental-math use of the binomial square formula.

Submit

15) A landscaper uses (a + b)² to find the area of a square with side (50 + 3).

2500 + 300 + 9

2500 + 30 + 9

2500 + 2(3) + 9

2500 + 150 + 9

2500 + 300 + 9, because (50 + 3)² = 2500 + 2·50·3 + 9 = 2500 + 300 + 9 = 2809, demonstrating the real-world usefulness of the binomial expansion in calculating area quickly.

Explanation

2500 + 300 + 9, because (50 + 3)² = 2500 + 2·50·3 + 9 = 2500 + 300 + 9 = 2809, demonstrating the real-world usefulness of the binomial expansion in calculating area quickly.

Submit

16) The volume of a cube changes when each side doubles. If the original edge is x, new volume = _____.

Because doubling the edge gives new side = 2x, and volume = (2x)³ = 8x³, showing cubic growth under linear scaling.

Explanation

Because doubling the edge gives new side = 2x, and volume = (2x)³ = 8x³, showing cubic growth under linear scaling.

Submit

17) Expand (a + b + c)².

A² + b² + c² + 2ab + 2bc + 2ca

A² + b² + c² + ab + bc + ca

A² + b² + c² + 3ab + 3bc + 3ca

A² + b² + c²

Every variable is squared once and each pairwise product appears twice since ab and ba both appear when expanding, giving the complete symmetric second-power expansion.

Explanation

Every variable is squared once and each pairwise product appears twice since ab and ba both appear when expanding, giving the complete symmetric second-power expansion.

Submit

18) Simplify (x³ + 8).

(x − 2)(x² + 2x + 4)

(x − 2)(x² − 2x + 4)

(x + 2)(x² + 2x + 4)

(x + 2)(x² − 2x + 4)

(x + 2)(x² − 2x + 4), because this is the sum-of-cubes identity a³ + b³ = (a + b)(a² − ab + b²), and substituting a = x and b = 2 gives (x + 2)(x² − 2x + 4).

Explanation

(x + 2)(x² − 2x + 4), because this is the sum-of-cubes identity a³ + b³ = (a + b)(a² − ab + b²), and substituting a = x and b = 2 gives (x + 2)(x² − 2x + 4).

Submit

19) Expand (a + b + c)³.

A³ + b³ + c³ + 3a²b + 3ab²

A³ + b³ + c³ + 3ab² + 3b²c

A³ + b³ + c³ + 3a²b + 3ab² + 3b²c + 3bc² + 3c²a + 3ca² + 6abc

A³ + b³ + c³ + 6abc

a³ + b³ + c³ + 3a²b + 3ab² + 3b²c + 3bc² + 3c²a + 3ca² + 6abc, because each variable appears cubed once, every ordered pair generates a term with coefficient 3, and the triple product abc appears with coefficient 6 due to symmetry in all permutations.

Explanation

Submit

20) Simplify (x² − y²)².

X⁴ + 2x²y² + y⁴

X⁴ − 2x²y² + y⁴

X⁴ + y⁴

X⁴ − y⁴

x⁴ − 2x²y² + y⁴, because applying the square identity (a − b)² = a² − 2ab + b² with a = x² and b = y² yields x⁴ − 2x²y² + y⁴, showing how nested exponents expand via standard binomial rules.

Explanation

Submit