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 472 mentally using (a−b)2=a2−2ab+b2 with a=50,b=3.

Example 2

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

Example 3

medium
Factor 25x2−16.

Example 4

medium
Expand (2x+3)2.

Example 5

easy
Factor a2−b2.

Example 6

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

Example 7

easy
Factor x2−25.

Example 8

challenge
Show that (a2+b2)(c2+d2)=(ac+bd)2+(ad−bc)2.

Example 9

easy
Compute (x+3)2.

Example 10

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

Example 11

medium
Factor x3+27 as a sum of cubes.

Example 12

medium
Factor x4−16 completely.

Example 13

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

Example 14

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

Example 15

easy
Expand (a−b)2.

Example 16

challenge
If x−1x=3, find x3−1x3.

Example 17

medium
Factor 9−6x+x2.

Example 18

medium
Expand (x−3)3.

Example 19

medium
If a+b=7 and ab=10, find a2+b2.

Example 20

easy
Compute (x+1)(x−1).
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