Use structure to rewrite expressions · lesson

Difference of squares

💡 a² − b² always factors as (a − b)(a + b).

Look for two perfect squares with a minus sign between them. Then a² − b² = (a − b)(a + b).

Find a and b by taking the square root of each term. For 25x², a is 5x.

A sum of squares, like x² + 9, does not factor this way.

🧠 Let's work it out

Factor 25x² − 36.

25x² = (5x)², so a = 5x.
36 = 6², so b = 6.
Use (a − b)(a + b): (5x − 6)(5x + 6).
Check: 25x² + 30x − 30x − 36 = 25x² − 36.

What do you think comes next?

difference of squares: one perfect square minus another, like x² − 36
factor: rewrite as a product

🔗 Khan Academy: Factoring differences of squares · Khan Academy ↗

Perfect square trinomials

💡 If the first and last terms are squares and the middle is twice their product, the trinomial is a binomial squared.

a² + 2ab + b² = (a + b)², and a² − 2ab + b² = (a − b)².

Check: are the first and last terms perfect squares? Is the middle term 2 times their roots multiplied? The sign of the middle term goes in the binomial.

🧠 Let's work it out

Factor x² − 12x + 36.

The ends are squares: x² = (x)² and 36 = 6².
Check the middle: 2 · x · 6 = 12x, and the middle is −12x.
So it is (x − 6)².
Another example: 9x² + 6x + 1 = (3x + 1)².

What do you think comes next?

perfect square trinomial: a three-term expression equal to a binomial squared

Greatest common factor with powers

💡 Pull out the largest number and the lowest power of x that divide every term.

The GCF of terms with powers is the greatest common number times x to the smallest exponent.

Divide each term by the GCF to see what stays inside.

🧠 Let's work it out

Factor 12x⁵ − 8x³.

The GCF of 12 and 8 is 4.
The lowest power is x³, so the GCF is 4x³.
Divide each term: 12x⁵ ÷ 4x³ = 3x² and 8x³ ÷ 4x³ = 2.
Write it: 4x³(3x² − 2).

What do you think comes next?

greatest common factor: the biggest factor that divides every term

Factoring completely

💡 After one factoring step, check each factor again; keep going until nothing more factors.

A factor can sometimes be factored again. x⁴ is a square, (x²)², so x⁴ − 81 is a difference of squares.

After the first step, check each piece. A difference of squares factors again. A sum of squares does not.

🧠 Let's work it out

Factor x⁴ − 81 completely.

See it as squares: (x²)² − 9² = (x² − 9)(x² + 9).
The factor x² − 9 is a difference of squares: (x − 3)(x + 3).
The factor x² + 9 doesn't factor, so the answer is (x² + 9)(x − 3)(x + 3).

What do you think comes next?