Factoring Higher Degree Polynomials

1
Factoring Higher Degree Polynomials

• And the Factor Theorem (7.6)

2
POD

• Factor these polynomial functions by grouping.
  In this case, there is no middle term to split
  we just group the first pair and last pair.

3
When you cant do this

• there are other strategies you can use.
• You need to find one factor to start. Synthetic
  division will help. Well start with this
  equationthe leading coefficient is 1, so it is a
  simple example.
• What is the constant? What are the factors of
  that constant? Be sure to include negative
  factors as well.

4
Using synthetic division

• By tables, working on the board, divide one of
  those factors into the polynomial using synthetic
  division. Which number(s) give a remainder of 0?

5
Using synthetic division

• By tables, divide one of those factors into the
  polynomial using synthetic division. Which
  number(s) give a remainder of 0?
• Lets use 1. If we have a remainder of 0 when we
  divide 1 into the polynomial, what does that mean
  about (x-1)? What is the quotient?

6
Using synthetic division
Can we factor the quotient? If so, what is it?
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Using synthetic division
Once we factor that quadratic quotient, we have
factored the entire polynomial. And we did
it by starting with one factor.
8
The Factor Theorem

• Lets go back to that first step, when we found a
  factor because the remainder was 0.
• The factor is (x-1). What is f(1)?

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The Factor Theorem

• f(1) 0
• Thats the Factor Theorem
• The (x a) is a factor of f(x), if and only if
  f(a) 0.
• In our polynomial, f(1) 0, so (x 1) is a
  factor.

10
Expanding the tool

• What about when the leading coefficient is not 1?
  Lets look at another example. This time, the
  leading coefficient is 2.
• What are the factors of 2? Of 3? Be sure to
  include positive and negative factors.

11
Expanding the tool

• We use synthetic division again to find a
  remainder of 0, but modify what we plug in to
  divide.
• The divisors are fractions
• factors of constant/factors of leading
  coefficient.
• In this case, well divide by . Do it
  again by tables, working on the board, to save
  some time.

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Expanding the tool

• Wow, a fraction.
• Lets tidy it up some. We want integers.
• Ill help out by saying that the quadratic factor
  cannot be factored any more, so were done.

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Expanding the tool

• Now, graph the polynomial. Where does it cross
  the x-axis?
• How does this match our factoring?
• When youre stuck, another tool in the tool box
  is to graph and find an x-intercept. That can
  give you a zero, which means having a factor to
  start with.

14
A caution

• If you divide by all the possible numbers, and
  dont have any remainder of 0, then there are no
  rational factors the polynomial cannot be
  factored over the set of rational numbers.

15
Finally

• in answer to a question last year, here are the
  patterns for factoring sums and differences of
  powers greater than 3. What patterns do you see?
  Could you factor a9 b9?