Section 0.5 Factoring Polynomials
You need to know how to: Factor out a common monomial, factor by grouping, factor the difference
of two squares, and factor trinomials (either by guess and check method or by grouping).
Factoring Out a Common Monomial
Ex: Factor 2x2 y + 6y.
What factors do these two terms have in common? 2x2 y = 2 · x · x · y and 6y = 2 · 3 · y. Both have
2y so we can factor out this:
2y(x2 + 3)
Factoring by Grouping
There may not be a factor common to all terms in the polynomial. However, there may be a factor
common to some of the terms. We can group the terms with a factor in common together and then
factor.
Ex: Factor ax + bx + a + b.
The first two terms both have an x. So we can factor out x. The third and fourth term have no
common factor, so we factor out a 1:
x(a + b) + 1(a + b).
Notice now that we have two terms, x(a + b) and 1(a + b). Each of these has a term in common,
(a + b). We can factor out (a + b):
(a + b)(x + 1).
Factoring the Difference of Two Squares
We use this method when we notice the binomial has a perfect square minus another perfect square.
Factoring the Difference of Two Squares: x2 − y 2 = (x + y)(x − y).
Ex: Factor 81y 2 − 36.
Each term is a perfect square:
81y 2 − 36 = (9y)2 − (6)2
Using the formula above:
= (9y − 6)(9y + 6).
Could have factored out a common monomial, 9. Then 81y 2 −36 = 9(9y 2 −4) = 9(3y −2)(3y +2)
NOTE: Because we are limited to integer coefficients, the sum of two squares cannot be factored.
Ex: x2 + y 2 is a prime polynomial.
Factoring Trinomials
There is a special formula for trinomial squares. Memorize these:
x2 + 2xy + y 2 = (x + y)(x + y) = (x + y)2
x2 − 2xy + y 2 = (x − y)(x − y) = (x − y)2
Ex: Factor x2 + 6x + 9.
x2 + 6x + 9 = x2 + 2 · 3x + 32 . Use the first formula:
x2 + 6x + 9 = (x + 3)2
Strategy for Factoring a General Polynomial with Integer Coefficients:
1. Write the trinomial in descending powers of one variable.
2. Factor out any greatest common factor, including −1 if it is necessary to make the coefficient
of the first term positive.
3. When the sign of the first term in a trinomial is + and the sign of the third term is +, the
sign between the terms of each binomial is the same as the sign of the middle term.
4. When the sign of the third term is -, one of the signs between the terms of the binomial factors
is + and the other is -.
5. Try various combinations of first terms and last terms until you find one that works. If none
work, the polynomial is prime.
6. Check the factorization by multiplication.
Ex: Factor x2 − 4x − 12.
Notice that both signs are −. Factor 12: Either 1 · 12 or 2 · 6 or 3 · 4. Since the third sign is −, we
know the signs are different. So, which factor pair subtracts to give us −4? This can be obtained
by −6 and +2. So we can factor:
x2 − 4x − 12 = (x − 6)(x + 2).
Ex: Factor 10x2 − 17xy + 6y 2 .
Factor the 1st and 3rd terms: 10 = 1 · 10 or 2 · 5 and 6 = 1 · 6 or 2 · 3. The third sign is + so we need
a combination that adds up to −17. (Both signs in the factors will be negative.) 2 · 2 + 5 · 3 6= 17.
2 · 3 + 5 · 2 6= 17 But 2 · 6 + 5 · 1 = 17. So we can factor:
10x2 − 17xy + 6y 2 = (2x − 1y)(5x − 6y). Notice how we paired the factors in the parenthesis so
that 2 is multiplied by 6 to get the middle term, etc. Always foil your answer to make sure you did
not make a mistake.
Ex: Let’s do the previous example by grouping:
10x2 − 17xy + 6y 2 . The ac product is 10 · 6 = 60. We need two factors of 60 whose sum is b= −17.
60 = 1 · 60 or 2 · 30 or 3 · 20 or 4 · 15 or 5 · 12 or 6 · 10. We want 5 · 12. So we write:
10x2 − 17xy + 6y 2 = 10x2 − 5xy − 12xy + 6y 2 . Now we factor by grouping:
10x2 − 17xy + 6y 2 = 10x2 − 5xy − 12xy + 6y 2 = 5x(2x − y) − 6y(2x − y) = (5x − 6y)(2x − y).
NOTE: −20and + 3 also give −17 but this combination does not work. We need the signs to be the
same.
Identifying Factoring Problem Types:
1. Factor out all common monomials.
2. If an expression has two terms, check whether the problem type is
• The difference of two squares: x2 − y 2 = (x + y)(x − y)
• The sum of two cubes: x3 + y 3 = (x + y)(x2 − xy + y 2 )
• The difference of two cubes: x3 − y 3 = (x − y)(x2 + xy + y 2 )
3. If an expression has 3 terms, try to factor it as a trinomial.
4. If an expression has 4 or more terms, try factoring by grouping.
5. Continue until each individual factor is prime.
6. Check results by multiplying.