1. Precalc – 2.3, 2.5 Finding zeros

2. DO NOW Create a function that COULD represent these graphs:

3. Algebra  Graph What do I know already about this guy? f(x) = -x 5 + x 4 + 2x 3 – 3x What more do I need to know?

4. Standard  Linear Factors Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n>0, then f has at least one zero *in the complex number system. Linear Factorization Theorem If f(x) is a polynomial of degree n, where n>0, then n has precisely n linear factors and f(x) = a(x-b)(x-c)(x-d) … (x-n) *but the factors MIGHT be complex!

5. To find the zeros Rational Zero Test If the polynomial f(x) has integer coefficients, every rational zero has the form p/q, where p and q have no common factors and: p = a factor of the constant term q = a factor of the leading coefficient In other words, possible rational zeros = Factors of constant term Factors of leading coef.

6. *note: not ever possible zero actually is a zero. plug it into the equation to check.

7. EXAMPLE. What are the rational zeros of… f(x) = x 3 + x + 1

8. Finding more zeros Once we can find one of the zeros, we can use PoLYNOmiAL DIVIsioN to find the rest

9. POLYNOMIAL DIVISION You can divide polynomials pretty much just like you can divide numbers It only seems more difficult because you haven’t done it five million times, like you have with normal numbers

10. Example x 2 + 3x + 5 / x + 1

11. SYNTHETIC DIVISION Why? Because it’s way easier then long division. (Let’s never do that again.)

12. SYNTHETIC DIVISION SHORTCUT! I love it. NOTE: only works for divisors like (x-k) (x+k) is okay too

13. Example x 2 + 3x + 5 / x + 1

14. Back to roots… I can use division to check the roots from the rational root theorem If you get a remainder, then the root doesn’t really WORK – throw it out!

15. Standard  Factored Given a polynomial in factored form: 1. Use the RZT to find possible zeros. 2. Check them, using synthetic division. 3. Using synthetic division, Rewrite the polynomial as a product of polynomials of smaller degrees. 4. If you have factors that are quadratic or higher, keep factoring! Quad: factor as you learned last week Higher: start this process over – hurray!

16. FACTOR ME. f(x) = 2x 4 + 7x 3 – 4x 2 – 27x – 18 f(x) = 2x 3 – x 2 – 13x – 6