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# Factoring Trinomials Using Patterns

If we recognize that a given polynomial follows a certain pattern, we can use that pattern to factor the polynomial.

Pattern â€” To Factor a Perfect Square Trinomial
 Perfect Square Trinomial: a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

Example 1

Factor: 25y2 - 10x2y + x4

Solution

Check the form of 25y2 - 10x2y + x4:

â€¢ there are three terms;

â€¢ the first term is a perfect square: 25y2 = (5y)2;

â€¢ the third term is a perfect square: x4 = (x2)2; and

â€¢ the second term is twice the product of the factors being squared: 10x2y = 2(5y)(x2).

 So, the given polynomial is a perfect square trinomial: Substitute 5y for a and x2 for b: a2 - 2ab + b2  (5y)2 - 2(5y)(x2) + (x2)2 = (a + b)2= (5y - x2)2

Thus, the factorization is (5y - x2)2.

You can multiply to check the factorization.

Note:

We can use FOIL to check the factorization:

 (5y - x2)2 = (5y - x2)(5y - x2) = 25y2 - 5x2y - 5x2y + x4 = 25y2 - 10x2y + x4

The factorization checks.

Pattern â€” To Factor the Difference of Two Perfect Squares

Difference of Two Perfect Squares: a2 - b2 = (a + b)(a - b)

Example 2

Factor: 16w2 - 81y6.

Solution

Check the form of 16w2 - 81y6.

â€¢ there are two terms;

â€¢ the first term is a perfect square: 16w2 = (4w)2;

â€¢ the second term is a perfect square: 81y6 = (9y3)2.

 So, the given polynomial is a difference of two squares: Substitute 4w for a and 9y3 for b: a2 - b2  (4w)2 - (9y3)2 = (a + b)(a - b)= (4w + 9y3)(4w - 9y3)

Thus, the factorization is (4w + 9y3)(4w - 9y3). You can multiply to check the factorization.

Note:

The polynomial 16w2 + 81y6 is not a difference of two squares because the terms are added rather than subtracted.

The polynomial 16w2 + 81y6 cannot be factored using integers.

We can also use patterns to factor the sum or difference of two cubes.

Patterns â€” To Factor the Sum or Difference of Two Cubes

Sum of Two Cubes:  a3 + b3 = (a + b)(a2 - ab + b2)

Difference of Two Cubes: a3 - b3 = (a - b)(a2 + ab + b2)

Example 3

Factor: 8y3 - 27x6

Solution

Check the form of 8y3 - 27x6:

â€¢ there are two terms;

â€¢ the first term is a perfect cube: 8y3 = (2y)3;

â€¢ the second term is a perfect cube: 27x6 = (3x2)3.

So, the polynomial is a difference of two cubes.

 Substitute 2y for a and 3x2 for b:   Simplify. a3 - b3 (2y)3 - (3x2)3 = (a - b) (a2 + ab + b2) = (2y - 3x2)[(2y)2 + (2y)(3x2) + (3x2)2] = (2y - 3x2)(4y2 + 6x2y + 9x4)

Thus, the factorization is (2y - 3x2)(4y2 + 6x2y + 9x4).

You can multiply to check the factorization.