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Math 35
5.6 "Factoring Trinomials"
Objectives:
* Factor perfect-square trinomials.
* Factor trinomials of the forms x2 + bx + c and ax2 + bx + c:
* Use substitution to factor trinomials.
* Use the grouping method to factor trinomials.
We will now discuss techniques for factoring trinomials. These techniques are based on the fact that the product of two
binomials is often a trinomial.
Factor Perfect-Square Trinomials
Factoring Perfect-Square Trinomials:
a2 + 2ab + b2 =
a2
2ab + b2 =
Example 1: (Factoring perfect-square trinomials)
Factor the following trinomials.
a) x2 + 10x + 25
b) 4x4
12x2 y 2 + 9y 4
Factor Trinomials of the Form x2+bx+c
We will consider any trinomials of the form x2 + bx + c where the coe¢ cient of the square term is 1 (leading coe¢ cient)
Factoring Trinomials Whose Leading Coe¢ cients is 1:
To factor a trinomial of the form x2 + bx + c, …nd two numbers whose product is c and whose sum is b
1.
If c is positive, the numbers have the same sign.
2: If c is negative, the numbers have di¤erent signs.
Then write the trinomial as a product of two binomials.
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
5.6
Example 2: (Factoring trinomials of the form x2 + bx + c)
Factor the following trinomials.
a) x2 + 11x + 24
b) a2 + 13a + 40
c) t4
d) 2x2 y 2 + 4xy 3
8t2 + 12
30y 4
Factor Trinomials of the Form ax2+bx+c
Factoring Trinomials by Grouping (Leading Coe¢ cients Other Than 1)
To factor a trinomial by grouping
1.
Factor out any GCF.
2.
Identify a; b; and c, and …nd the key number ac:
3.
Find two integers whose product is the key number and whose sum is b:
4.
Express the middle term, bx; as the sum (or di¤erence) of two terms.
5.
Enter the two numbers found in step 2 as coe¢ cients of x in the form shown below.
6.
Factor the equivalent four-term polynomial by grouping.
Example 3: (Factoring trinomials of the form ax2 + bx + c)
Factor the following trinomials.
a) 5x2 + 7x + 2
b) 3p2
Page: 2
4p
4
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
c)
6x2
e) 5a2 b
15x
5.6
d) 4 + 11a2 + 6a4
6
f) 15a2 + 17ab
8ab + 2b
4b2
Use Substitution to Factor Trinomials
For more complicated expressions, especially those involving a quantity within parentheses, a substitution sometimes
helps to simplify the factoring process.
Example 4: (Using substitution to factor trinomials)
Factor the following polynomials using substitution.
2
a) (x + y) + 7 (x + y) + 12
b) (a + b)
Page: 3
2
3 (a + b)
10
Notes by Bibiana Lopez