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Creating Polynomials Given the Zeros.
What do we already know about polynomial functions?
They are either ODD
functions
They are either EVEN
functions
Lineary = 4x - 5
Cubicy = 4x3 - 5
Fifth Powery = 4x5 –x + 5
Quadraticsy = 4x2 - 5
Quarticsy = 4x4 - 5
Quadraticsy = 4x2 - 5
We 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**
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
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.
i and 2 :of solutionsget wouldWe
0 )1)(8(4x
: withup end and polynomial afactor we22
x
If
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the
given zeros.
-1, 2, 4Step 1: Turn the zeros into factors.
(x+1)(x- 2)(x- 4)Step 2: Multiply the factors together.
x3 - 5x2 +2x + 8Step 3: Name it!
f(x) =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.
Step 1: Turn the zeros into factors.
3 ,2i
Must remember that “i”s and roots come in pairs.
3,3 ,2,2 ii
)3)(3-x)(2)(2( xixix
Step 2: Multiply factors.)3-x)(4( 22 x 12)( 24 xxxf
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.
3-2 ,2 i
Must remember that “i”s and roots come in pairs.
32,3-2 ,2- ,2 ii
)32)(32-x)(2)(2( xixix
Step 2: Multiply factors.
)32)(32-x)(2)(2( xixix
x 2 ix
2
-i
x2 2x ix2i42x
-ix -2i -i2 1xxx
x(x2+ 4x + 5)
x -2 3
x
-2
3
x2 -2x
4-2x
-3
3x
3x 32
32
xxx
x(x2- 4x + 1)
x2
-4x
1
x2+ 4x + 5x4 4x3
-3f(x) = x4-10x2 -16x + 5
-4x3
x2
-16x2
5x2
-20x
54x