LEMMA

Polynomials & quadratics

Special polynomial products

ALEKS placement

Three patterns cover most expansions, and recognizing them saves the work.

What this covers

(a+b)2=a2+2ab+b2,(a−b)2=a2−2ab+b2,(a+b)(a−b)=a2−b2
  • Check whether the factors are identical (a square) or conjugates (a sum and a difference).
  • For a square, the middle term is twice the product of the two parts.
  • For conjugates the middle terms cancel, leaving only the difference of squares.
  • Square the whole term, coefficient included.

Worked example

Worked example

Expand (3x - 5)².

  1. (3x)2−2(3x)(5)+52First squared, twice the product, last squared.
  2. 9x2−30x+25Simplify each piece.
Another worked example

Expand (x − 5)².

  1. (x−5)(x−5)=x2−5x−5x+25Squaring repeats the entire binomial, including its negative term.
  2. (x−5)2=x2−10x+25The cross-products reinforce; the product of two negatives is positive.

A common mistake

A mistake Lemma catches

From (3x−5)2

9x2+259x2−30x+25

“You squared 3x and -5 separately, but (3x - 5)² is (3x - 5)(3x - 5) — multiplying it out also gives the middle term -30x.”

Try one

Sample problem

Expand.

(x+2)2

Practice special products free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

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