LEMMA

Polynomials & quadratics

Difference of squares

ALEKS placement

Two squares with subtraction become matching factors with opposite signs.

What this covers

a2−b2=(a+b)(a−b)
  • Confirm both terms are perfect squares and the operation is subtraction.
  • Take the square root of each term.
  • Write the sum times the difference.
  • Check whether either factor is itself a difference of squares.

Worked example

Worked example

Factor 9x² − 16 over the reals

  1. 9x2=(3x)2The first term is the square of the entire quantity three times the variable.
  2. 16=42The second term is also a square. The subtraction sign between them is essential.
  3. 9x2−16=(3x−4)(3x+4)Use one group with subtraction and one with addition. These matching groups are called conjugates.
  4. (3x−4)(3x+4)=9x2+12x−12x−16Distribute to check all four products. The two middle terms have opposite signs.
  5. 9x2+12x−12x−16=9x2−16The middle terms cancel. The remaining expression is exactly the original difference of squares.
Another worked example

Factor 16x⁴ − 81 over the reals

  1. 16x4−81=(4x2)2−92Identify the whole squared quantities. The variable part can itself contain a power.
  2. (4x2+9)(4x2−9)Apply the difference-of-squares pattern once. One of the resulting factors has another subtraction of squares.
  3. 4x2−9=(2x+3)(2x−3)Factor the remaining difference using its own two squared quantities.
  4. (4x2+9)(2x+3)(2x−3)Keep the other factor in the final product. That sum has no real linear factors.

A common mistake

A mistake Lemma catches

From 9x2

9x2=(9x)29x2=(3x)2

“9x² is the square of 3x, because (3x)² = 3² · x² = 9x². Writing (9x)² squares the 9 again: (9x)² = 81x², which is not 9x².”

Try one

Sample problem

Factor completely.

x2−100

Practice difference of squares 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

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