1. Creating Polynomials Given the Zeros.

6. What do we already know about polynomial functions?
They are either ODD functions
They are either EVEN
functions
Linear
y = 4x - 5
Cubic
y = 4x3 - 5
Quadratics
y = 4x2 - 5
Quadratics
y = 4x2 - 5
Fifth Power
y = 4x5 –x + 5
Quartics
y = 4x4 - 5

7. We know that factoring and then solving those factors set equal to zero allows us to find possible x intercepts.
TOOLS WE’VE USED
Long Division (works on all factors of any degree)
Factoring
GCF
Quadratic Formula
(x + )(x + )
Synthetic Division (works only with factors of degree 1)
The “6” step
Grouping
Cubic**
p/q

8. Zeros are x-intercepts if they are real
We know that solutions of polynomial functions can be rational, irrational or imaginary.
X intercepts are real.
Zeros are x-intercepts if they are real
Zeros are solutions that let the polynomial equal 0

9. We have seen that imaginaries and square roots come in pairs ( + or -).
So we could CREATE a polynomial if we were given the polynomial’s zeros.

10. -1, 2, 4 (x+1)(x- 2)(x- 4) f(x) =x3 - 5x2 +2x + 8 x3 - 5x2 +2x + 8
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros.
-1, 2, 4
Step 1: Turn the zeros into factors.
(x+1)(x- 2)(x- 4)
Step 2: Multiply the factors together.
f(x) =x3 - 5x2 +2x + 8
x3 - 5x2 +2x + 8
Step 3: Name it!

11. Step 1: Turn the zeros into factors.
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros.
Must remember that “i”s and roots come in pairs.
Step 1: Turn the zeros into factors.
Step 2: Multiply factors.

12. Step 1: Turn the zeros into factors.
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros.
Step 1: Turn the zeros into factors.
Must remember that “i”s and roots come in pairs.
Step 2: Multiply factors.

13. x x x x x x x x (x2+ 4x + 1) (x2+ 4x + 3) x2 2x x2 2x ix 4 2i 2x 2x 4-1 -3
(x2+ 4x + 1)
(x2+ 4x + 3)

14. x2+ 4x + 3 f(x) = x4+ 8x3 + 20x2 +16x + 3 x4 4x3 3x2 4x3 16x2 12x x2-3