Practice Algebraic Pattern in Math

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

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Quick Recap

A recognizable, recurring algebraic structure such as a2โˆ’b2a^2 - b^2 or (a+b)2(a+b)^2 that can be applied systematically.

a2โˆ’b2a^2 - b^2 always factors to (a+b)(aโˆ’b)(a+b)(a-b) โ€” recognize the pattern once and apply it everywhere.

Showing a random 20 of 50 problems.

Example 1

hard
Simplify a2โˆ’b2aโˆ’b\dfrac{a^2 - b^2}{a - b} for aโ‰ ba \ne b.

Example 2

medium
A trinomial x2+bx+9x^2 + bx + 9 is a perfect square. Find all values of bb.

Example 3

easy
Factor x2+7x+12x^2 + 7x + 12 by finding the additive/multiplicative pattern.

Example 4

medium
The sequence 2,5,10,17,26,โ€ฆ2, 5, 10, 17, 26, \dots follows what pattern? Give a formula for ana_n.

Example 5

medium
Factor x3+125x^3 + 125 using the sum-of-cubes pattern.

Example 6

challenge
Recognize the pattern in (n0)+(n1)+โ‹ฏ+(nn)\binom{n}{0}+\binom{n}{1}+\dots+\binom{n}{n} and prove the closed form using the binomial theorem.

Example 7

medium
Use a pattern to compute 1032โˆ’972103^2 - 97^2.

Example 8

easy
Recognize the pattern: 1,4,9,16,โ€ฆ1, 4, 9, 16, \dots. What is the nn-th term?

Example 9

hard
Recognize and factor: x4+4x2+4โˆ’9x2x^4 + 4x^2 + 4 - 9x^2.

Example 10

easy
Is x2+16x^2 + 16 factorable over the reals as a difference of squares?

Example 11

medium
Factor 4x2โˆ’254x^2 - 25 by recognizing the disguised pattern.

Example 12

medium
Factor 9x2+30x+259x^2 + 30x + 25.

Example 13

challenge
For which integer nn is n4+4n^4 + 4 factorable over the integers as a product of two quadratics? Use the Sophie Germain pattern.

Example 14

medium
Use a pattern to compute 99ร—10199 \times 101 without long multiplication.

Example 15

easy
Is x2+4x^2 + 4 a difference of squares?

Example 16

medium
The pattern 11โ‹…2+12โ‹…3+13โ‹…4\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4} telescopes. Recognize the term pattern and sum it.

Example 17

challenge
Find a closed form for the telescoping sum โˆ‘k=1n1k(k+1)\sum_{k=1}^{n} \dfrac{1}{k(k+1)}.

Example 18

easy
Identify the pattern and factor: x2โˆ’2x+1x^2 - 2x + 1.

Example 19

medium
Find bb so that x2+bx+36x^2 + bx + 36 is a perfect square.

Example 20

easy
Factor x2โˆ’9x^2 - 9 by recognizing the pattern.
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