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Factoring Higher Degree Polynomials

Factoring Higher Degree Polynomials. And the Factor Theorem (7.6). 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. When you can’t do this…. …there are other strategies you can use.

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Factoring Higher Degree Polynomials

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  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 can’t do this… …there are other strategies you can use. You need to find one factor to start. Synthetic division will help. We’ll start with this equation—the 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? Let’s 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?

  7. 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 Let’s 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)?

  9. The Factor Theorem f(1) = 0 That’s 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? Let’s 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, we’ll divide by . Do it again by tables, working on the board, to save some time.

  12. Expanding the tool Wow, a fraction. Let’s tidy it up some. We want integers. I’ll help out by saying that the quadratic factor cannot be factored any more, so we’re done.

  13. Expanding the tool Now, graph the polynomial. Where does it cross the x-axis? How does this match our factoring? When you’re 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 don’t 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?

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