Practice Algebraic Identities in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Algebraic identities are equalities true for all permitted values of their variables.

Identities are always-true shortcuts — no matter what values you substitute, both sides will always be equal.

Showing a random 20 of 50 problems.

Example 1

medium
Compute 47247^2 mentally using (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 with a=50,b=3a = 50, b = 3.

Example 2

challenge
Prove (a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)(a+b+c)^3 = a^3 + b^3 + c^3 + 3(a+b)(b+c)(c+a).

Example 3

medium
Factor 25x2−1625x^2 - 16.

Example 4

medium
Expand (2x+3)2(2x + 3)^2.

Example 5

easy
Factor a2−b2a^2 - b^2.

Example 6

medium
Factor x2+10x+25x^2 + 10x + 25 using a square identity.

Example 7

easy
Factor x2−25x^2 - 25.

Example 8

challenge
Show that (a2+b2)(c2+d2)=(ac+bd)2+(ad−bc)2(a^2+b^2)(c^2+d^2) = (ac+bd)^2 + (ad-bc)^2.

Example 9

easy
Compute (x+3)2(x+3)^2.

Example 10

challenge
Show that n4+4n^4 + 4 factors over the integers for n≥1n \ge 1. Factor it.

Example 11

medium
Factor x3+27x^3 + 27 as a sum of cubes.

Example 12

medium
Factor x4−16x^4 - 16 completely.

Example 13

medium
Verify whether a2+b2=(a+b)2a^2 + b^2 = (a+b)^2 for a=1,b=2a=1, b=2.

Example 14

hard
Factor x3+8x^3 + 8 using the sum of cubes identity.

Example 15

easy
Expand (a−b)2(a-b)^2.

Example 16

challenge
If x−1x=3x - \dfrac{1}{x} = 3, find x3−1x3x^3 - \dfrac{1}{x^3}.

Example 17

medium
Factor 9−6x+x29 - 6x + x^2.

Example 18

medium
Expand (x−3)3(x-3)^3.

Example 19

medium
If a+b=7a+b = 7 and ab=10ab = 10, find a2+b2a^2 + b^2.

Example 20

easy
Compute (x+1)(x−1)(x+1)(x-1).
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