Exponents and Polynomials

Special Products

Three patterns to recognize: (x+a)^2 = x^2 + 2ax + a^2 (sum squared), (x-a)^2 = x^2 - 2ax + a^2 (difference squared), and (x+a)(x-a) = x^2 - a^2 (conjugate pair / difference of squares).

special-products Memorize the three patterns. For conjugate pairs, the middle terms cancel. For perfect squares, the middle term is 2ax.

Example

Apply common square and conjugate product patterns.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

(x+3)2(x+ 3 )^{ 2 }

Step 2 — Sum square

Use the sum-square pattern.

(x+3)2=x2+6x+9(x+ 3 )^{ 2 }= x^{2} + \hlmath{6x} + \hl{9}

Step 3 — Difference square

Use the difference-square pattern.

(x−2)2=x2−4x+4(x- 2 )^{ 2 }= x^{2} - \hlmath{4x} + \hl{4}

Step 4 — Conjugates

Use the difference-of-squares pattern.

(x+4)(x−4)=x2−16(x+ 4 )(x- 4 )= x^{2} - \hl{16}