Click here for a presentation on Factoring Other Trinomials
In the presentation you looked at trinomials, like 2x2 – 5x + 8. The difference between these and the trinomials you previously factored is that the coefficient of the square term is not one. When that is the case, use a different technique.
How to Factor a Trinomial of the Form ax2 + bx + c
Find the product of the first and last coefficients, (a)(c).
List the factors of the product and find their sums until the sum is the middle coefficient.
Rewrite the trinomial and factor by grouping.
Check using FOIL.
Example 1: Factoring a Trinomial
Factor 6x2 + 11x + 4.
Find the product of six and four: 6(4) = 24.
List the factors and sums.
Product of 24
Sum
1, 24
25
2, 12
14
3, 8
11
4, 6
10
Use three and eight since they give the sum of 11, the middle coefficient.
(3x – 2)(x + 1) = 3x2 + 3x – 2x – 2 = 3x2 + x – 2
F O I L
Example 3: Factoring by the GCF First
Sometimes a trinomial has a GCF which can be factored out. If we factor this first, the resulting trinomial will be a simpler one.
Factor 2x2 – 6x – 8.
2x2 – 6x – 8 = 2(x2 – 3x – 4)
Now we can factor x2 – 3x – 4 using the shorter method for simple trinomials.
Product of -4
Sum
-1, 4
3
1, -4
-3
We can stop here as 1 and -4 give the middle coefficient of -3.
Factor: 2(x + 1)(x – 4)
Check using FOIL.
2(x + 1)(x – 4) = 2(x2 + x – 4x – 4) = 2(x2 – 3x – 4) = 2x2 – 6x – 8
F O I L
Just as there are prime numbers, there are prime polynomials that cannot be factored.
Example 4: A Prime Trinomial
Factor 3x2 – 5x + 1.
Find the product of 3 and 1. 3(1) = 3.
List the factors and sums.
Product of 3
Sum
1, 3
4
-1, -3
-4
Since none of the factors will give a sum of negative five, the polynomial cannot be factored and is prime.
Example 5: Solving Equations by Factoring Trinomials
Solve 3x2 = 4x – 1.
Set it equal to zero.
Subtract 4x and add 1 3x2 – 4x + 1 = 0
Factor (3x – 1) (x – 1) = 0
Set = 0 3x – 1 = 0 or x – 1 = 0
Solve 3x = 0 + 1 x = 0 +1
x = 1/3 x = 1
Check:
Practice
Click to get a new problem. Factor the polynomial and then click to see the answer. Remember that your answer is still correct if you have the same binomials reversed.
In the following problems, factor the GCF, and then factor the resulting trinomial. Write the GCF and the two binomials together as the set of factors.
Trinomial
GCF(resulting trinomial)
Factors
6x2 – 14x – 12
2( 3x2 – 7x – 6)
2(3x + 2)(x – 3)
30x2 – 25x – 30
5(6x2 – 5x – 6)
5(3x + 2)(2x – 3)
3x2 + 24x + 45
3(x2 + 8x + 15)
3(x + 3)(x + 5)
4x2 + 16x – 20
4(x2 + 4x – 5)
4(x – 1)(x + 5)
4x2 + 8x – 4
4(x2 + 2x – 1)
4(x2 + 2x – 1)
since x2 + 2x – 1 is prime.
Solve each of the following equations by factoring.
{-2/3, 3}
2 cannot equal 0, so it does not yield a solution.
Section 5 Homework(10 points)
It’s time for your homework. Find the section homework link to submit it for a grade. You may take this only one time, so check your understanding of the material before completing your homework, and do your best.
Click here for a presentation on Factoring Other Trinomials
How to Factor a Trinomial of the Form ax2 + bx + c
Practice
Section 5 Homework(10 points)