1. There is a pattern for factoring trinomials of this form, when c
Factoring Patterns
There is a pattern for factoring trinomials of this form, when c
is positive
x² + bx + c

2. Trinomials have 3 terms x² + 9x +8
The first term is of degree two and so the term is called a quadratic term
The second term is called the linear term
The last term is called the constant
(it has no variable factor)
The trinomial itself is called a quadratic polynomial

3. Examples of trinomials in this form
x² + 9x + 8
r² + 10 r + 24
y² - 14y + 13
m² - 10m + 16
NOTE: coefficient of quadratic term is 1
constant term is positive
Notice that in each case the coefficient of the quadratic term is 1 and the constant term is positive.

4. To factor trinomials like x² + 8x + 16
List pairs of factors whose product equal the constant term
4 4
Find the pair of factors whose sum equals the coefficient of the linear term.
4 + 4 = 8

5. Factor x² + 8x + 16
(x + 4)(x + 4) or (x+4)²

6. y² + 20y + 36 Factors of 36 1 36 6 6 9 4 12 3 18 2 Sum of factors 37
Sum of factors 37 12 13 15 20

7. So factor y² + 20y + 36 (y + 18) ( y + 2)
Check by multiplying the binomials using FOIL
y² + 2y + 18y + 36
y² + 20y + 36

8. Factor x² - 12x + 20
Since the linear term is negative and the constant term is positive we must list the negative factors of 20.

9. So factor x² - 12x + 20
We know the only possible factors are –2 and –10 so we write
(x – 2)(x – 10)
Check by applying FOIL
x² -10x –2x + 20
x² - 12x + 20

10. Another Factoring Pattern
x² - ax – c
There is also a pattern for factoring trinomials of this form when c is negative

11. Trinomials with three terms
x² + 29x – 30
m² + 12m – 36
k² - 25 k – 54
g² -g – 2
Note: coefficient of quadratic term is 1
constant term is negative

12. To factor trinomials like x² + 7x - 18
List pairs of factors of -18
Sum of factors 3 -3 -7 7 17
-17

13. So factor x² + 7x - 18
Since the linear term is positive select factors which give a positive result when added. But remember, because the constant term is negative, one factor must be negative. Using the preceding factor list we can write
(x + 9) (x – 2)
Check using FOIL x² - 2x + 9x – 18
x² + 7x - 18