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# Factoring polynomials

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### Transcript

• 1. Factoring Polynomials
NCVPS
Summer 2010
• 2. Factoring is reversing the process of multiplying monomials, binomials or trinomials or any combination of these.
We will focus on factoring out monomials (we call these GCFs) from a polynomial or factoring trinomials into two binomials.
Ex: 3x(a – 3) or (x – 1)(x + 5)
| | /
monomial binomial binomials
What is factoring?
• 3. Sometimes a common factor is present.
Ex 1: 3x2 – 6x -9 has a common factor of 3:
3(x2 – 2x – 3)
Ex 2: x3-5x2+4x has a common x in all 3 terms:
x(x2 - 5x + 4)
Ex 3: 2x2a + 6xa+12a has a common factor of 2a:
2a(x2 + 3x + 6)
Step 1: Look for a GCF present
• 4. Factor first term into its products.
Example 1: x2– 2x – 3.
The first term is x2. It is factored as x*x.
Factor the last term into its products.
The last term is -3. It factors as 1(-3)
or (-1)(3)
Step 2: Factor first & last terms
• 5. Example 1: x2 – 2x – 3.
(x )(x )
Next, add the factors of -3:
x2– 2x – 3
(x -3)(x +1)
Example 1 continued
• 6. It is very important to check your factoring to make sure you got the signs in the correct place. Multiply them out again!
x2 - 2x – 3 = (x -3)(x +1)
F: x(x)
O: (-3)(1) = -3
I: -3x
L: 1x
Inner + Last terms = -2x
Check the work!
• 7. Checking the signs is very important!
x2 + 3x + 2 has all + signs. It factors with + signs: (x + 2)(x + 1)
x2 – 6x + 5 has a +5 but the middle term is -. So 5 must factor as (-5)(-1): (x - 5)(x – 1)
More examples on the next page 
Signs, signs, everywhere there’s signs!
• 8. X2 -2x – 8 has a -8 that factors as + and –
Here we work with factors and sums. We want a sum of the factors to be -2 (the middle term)
(-8)(1) = -8 sum the factors: -8+1 = -7
(8)(-1) = -8 sum the factors: 8 + (-1) = 7
(-4)(2) = -8 sum the factors: -4 + 2 = -2
This last one is the factorization we want!
(x – 4)(x + 2)
Keep on checking signs!
• 9. Find the factorization of x2 – 4x -12
-12 factors as (-3)(4) or (3)(-4) or (2)(-6) or (-2)(6) or
(-1)(12) or (1)(-12).
Which of these sums to -4?
-3 + 4 = 1
3 + (-4) = -1
2 + (-6) = -4 We have a winner!
x2 -4x – 12 is (x + 2)(x – 6)
Another example
• 10. x2 – 2x – 15
2x2 – x – 3
3x2-12x +9
Go to the next slide for the solutions.
You Try Problems
• 11. x2 – 2x – 15 = (x – 5)(x + 3)
2x2 – x – 3 = (2x - 3)(x +1)
3x2-12x +9 = 3(x2 – 4x + 3)
= 3(x – 3)(x -1)
You Try Problems