Download Section 6.3

Factoring Trinomials Whose Leading Coefficient is NOT "1" - Section 6.3
We now want to factor polynomials of the form ax 2
bx
c where a, b, and c are real numbers with a
0 or 1.
We will consider two methods: Trial-and-Error and Grouping or ac-Method.
Trial-and-Error Method:
First factor out any GCF other than 1 from the polynomial. Assume that the polynomial is now in the form
ax 2 bx c.
1. Find two first terms whose product is ax 2
x
x
ax 2
bx
c
2. Find the last two terms whose product is c
x
x
ax 2
bx
c
3. By trial and error, continue to perform steps 1 and 2 until the sum of the inner and outer products is bx
x
x
ax 2
bx
c
If no such combination exists, then the polynomial is prime.
Examples: Factor
1. 5x 2
14x
8
1
2. 6x 2
19x
7
Some helpful hints and suggestions for factoring ax 2
bx
c using trial-and-error:
1. If b is relatively small, avoid the larger factors of a.
2. If c is positive, the signs in both binomial factors must match the sign of b.
3. If the trinomial has no common factor, than no binomial factor can have a common factor.
4. Reversing the signs in the binomial factors changes the sign of the middle term bx.
Factor 6x 2
x
12 using trial-and error. Use some of the helpful hints and suggestions from above.
2
Factoring by the Grouping Method (or ac -Method):
To factor ax 2
bx
c
1. Calculate the product ac.
2. Find to factors, m and n, such that mn
ac and m
3. Rewrite the trinomial as ax 2
c and factor by grouping.
mx
nx
n
b.
Examples: Factor
1. 3x 2
x
2. 8a 2
3a
10
5
3
3. 12x 2
4. 3x 2
23x
5x
10
4
Factoring Trinomials in Two Variables:
5. Factor 3x 2
13xy
4y 2 by trial-and-error
4
6. Factor 3x 2
31xy
22y 2 by the grouping method
Factor the following polynomials COMPLETELY. Remember to start with factoring out the GCF.
7. 5y 4
13y 3
6y 2
5
8.
4x 3
9. 12a 2 b
2x 2
90x
21ab 2
6b 3
6