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Simple Trinomials as Products of Binomials
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The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
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Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Algebra
Solving Quadratic Equations Using the Square Root Property
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Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
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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 — 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:

• there are three terms;

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

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

• 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 — 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.

• there are two terms;

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

• 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 — 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:

• there are two terms;

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

• 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.

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