What do we already know about polynomial functions They are either ODD functions They are either EVEN functions Linear y 4x 5 Cubic y 4x 3 5 Fifth Power y 4x 5 x 5 Quadratics ID: 798101
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Slide1
Creating Polynomials
Given the Zeros.
Slide2What 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
Fifth Power
y = 4x5 –x + 5
Quadratics
y = 4x2 - 5
Quartics
y = 4x4 - 5
Quadratics
y = 4x
2 - 5
Slide3We know that factoring and then solving those factors set equal to zero allows us to find possible x intercepts.
TOOLS WE’VE USED
Factoring
Quadratic Formula
Long Division (works on all factors of any degree)
Synthetic Division (works only with factors of degree 1)
GCF
(x + )(x + )
The “6” step
Grouping
p/q
Cubic**
Slide4We 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
Slide5We 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.
Slide6Write 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.
x
3
- 5x2 +2x + 8
Step 3: Name it!
f(x) =x3 - 5x
2 +2x + 8
Slide7Write 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.
Slide8Write 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.
Slide9x 2
i
x
2
-
i
x
2
2x
ix
2i
4
2x
-ix
-2i
-i
2
1
x
x
x
x
(x
2
+ 4x +
5)
x
-2
x
-2
x
2
-2x
4
-2x
-3
x
x
x
x
(
x
2
-
4x
+ 1)
Slide10x
2
-4x
1
x
2
+ 4x +
5
x4
4x3
-3
f(x) = x4-10x2 -16x
+ 5
-4x3
x2
-16x2 5x2 -
20x5
4x