Download Sections 3.4 and 3.5 Factoring Polynomials in the form x2+bx+c

Sections 3.4 and 3.5
Factoring Polynomials in the form x2+bx+c.
Factor: 2x3 + 14x2 + 12x
2x3 = (2)(x)(x)(x)
14x2 = (2)(7)(x)(x)
12x = (2)(2)(3)(x)
Factor out the GCF
2x (x2 +7x +6)
GCF = 2x
expand through to check your answer
You may think that this expression is factored since the GCF of the terms in the brackets is 1
However, it is not completely factored. – Why?!!
To understand why, let’s first review (from grade 9) - multiplication of binomials. (expanding)
Method 1 : Distribution
a)
(x – 3)(x + 6)
= x (x+6)-3(x+6)
b)
(2x + 5)(x -2)
= 2x (x-2) +5 (x-2)
Method 2 : using the acronym FOIL (First, Outside, Inside and Last)
a) (2x+6)(x+8)
b) (x-5)(x-7)
The process of factoring is the opposite of expanding. Therefore each expanded trinomial
above can be factored into the product of two binomials. Notice the patterns and try to apply
it to x2 +7x +6
x2 + 7x + 6
What two numbers have a sum of +7 and have a product of +6?
Therefore, the factors of x2 + 7x + 6 = (x+1)(x+6)
So the question from above – Factor 2x3 + 14x2 + 12x
Would be completed factored to the form: 2x(x+1)(x+6)
Sum (7) Product (6)
1+6=7 1x6
-1 + (-6) = - 7 -1 x -6
2+3=5
2x3
-2 + (-3) = - 5 -2 x -3
Another method: Using Algebra Tiles
Using algebra tiles, we can determine the factors of x2 + 7x + 6.
x+1
x+6
By forming the tiles into a perfect rectangle, the side lengths represent the factors.
Therefore, x2 + 7x + 6=(x+6)(x+1)
Please Note: Not all quadratic trinomials (ax2+bx+c) can be factored to the product of two
binomials. It is your job to determine which ones can.
Example 2: Try to Factor x2 + 4x – 12
x2 + 4x – 12
What two numbers have a sum of +4 and have a product of -12?
-1 + 12 = 11
1 - 12 =-11
2 – 6 = -4
-2 + 6 = 4
Therefore, the factor of x2 + 4x - 12 = (x-2)(x+6)
Try these ! Factor each expression below.
a) x2 - 2x -8
c) x3 + x2 – 30x
Sum (4)
b) a2 - 7a -18
d) 3x3 – 15x2 – 72x
Product (-12)
-1 x 12
1 x -12
2 x -6
-2 x 6