Warm Up: Solve & Sketch the
graph:
f ( x)  x 3  x 2  2 x


0  x x2  x  2
0  xx  2x  1
x0
x 1
x  2
Graphing Polynomials &
Finding a Polynomial
Function
Equivalent Statements:

x = a is a zero of the function f.

x = a is a solution of the polynomial equation
f(x)=0.

(x-a) is a factor of the polynomial f(x)

(a, 0) is an x-intercept of the graph of f
Multiplicity
Repeated Zeros
For a polynomial function, a factor of x  a  , k  1,
k
yields a repeated zero x  a of multiplict y k.
1. If k is ODD , the graph CROSSES the x  axis
at x  a.
2. If k is EVEN , the graph TOUCHES the x  axis
at x  a.
f ( x)  2 x  2 x
4
Sketch the graph:
End behavior:
0  2 x  2 x
4

2

0  2 x x  1
2
2
0  2 x  x  1 x  1
2
x0
x  1
Multiplicity of 2.
Touches.
x 1
Through these
Points.
2
Multiplicity - repeated zero –
Means…………..

If it occurs an odd number of times, the
graph crosses the x-axis at the zero.

If it occurs an even number of times, the
graph will just touch the x-axis at the zero.
Sketch the graph:
0  3x x  2
x0 x2
2
3
Mult. of 3
Goes Through
Mult. Of 2
Touches
x y
1 3
f ( x)  3x ( x  2)
3
1st Term would be 3 x
End Behavior:
5
.
2
Sketch the graph:
f ( x)   x ( x  2)
2
1st Term would be
0   x x  2
x  0 x  2
2
2
Both have a multiplicity
of 2.
Just Touch!
x y
-1 -1
End Behavior:
 x4.
2
Can you write a
polynomial function if
you know the zeros?
Find a Polynomial function that has the given
zeros: 6, -3
x  6 x  3
x  6x  3
x  3 x  6 x  18
2
x  3x  18
2
Find a Polynomial function that has the given
zeros: 0, 3, -5
x  0 x  3 x  5
x x  3x  5
x
2

 3 x  x  5
x  5 x  3 x  15 x
3
2
2
x  2 x  15 x
3
2
Find a Polynomial function that has the given
zeros: 3, 2, -2, -1
x  3 x  2 x  2 x  1
x  3x  2x  2x  1
x
2

 5 x  6 x  3x  2
2

x 4  3 x 3  2 x 2  5 x 3  15 x 2  10 x  6 x 2  18 x  12
x  2 x  7 x  8 x  12
4
3
2
Find a Polynomial function that has the given
zeros: 5  2
and
5 2
x  5 2 x  5 2
x  5  2 x  5  2 
x 2  5 x  x 2  5 x  25  5 2  x 2  5 2  2
x  10 x  23
2
Long & Synthetic
Division
Use Long Division and use the result
to factor the polynomial completely.
x2
1st Step:
6 x3  19 x2  16 x  4
2nd Step:
x  2 6 x  7 x  2
x  22 x  13x  2
2
Divide using long division:


3x3  2 x 2  17 x  17   x  3
5
3x  7 x  4 
x3
3
2
x  3 3 x  2 x  17 x  17
3
2
3x  9 x
 7 x 2  17 x
2
 7 x  21x
2
4 x  17
4 x  12
Remainder
5
Divide using long division:


x3  64   x  4 
x  4 x  16
2
x  4 x 3  0 x 2  0 x  64
3
2
x  4x
2
4x  0x
4 x 2  16 x
x  4x
2
 4 x  16

16 x  64
16 x  64
0
You Try: Divide using long division


47
2

6 x  8 x  26 x  2
3
2
x  2 6 x  20 x  10 x  5
3
2
6 x  12 x
6 x3  20 x 2  10 x  5   x  2 
 8 x  10 x
2
 8 x  16 x
2
 26 x  5
 26 x  52
R :  47
Divide using long division

 
2 x 4  4 x3  5 x 2  3x  2  x 2  2 x  3

x 1
2x  1  2
x  2x  3
2
x  2 x  3 2 x  4 x  5 x  3x  2
2
4
3
2
Synthetic Division
Divide using synthetic division to find
the zeros:


x 2  10 x  21   x  3
31
1
 10 21
3 21
7
x  3x  7   0
x3
x7
0
Divide using synthetic division:

3

x 4  10 x 2  2 x  4   x  3
1
1
0
3
3
 10
2
9
3
1
1
4
3
1
1 
 3
2
x  3 x  3x  x  1 

x  3

You Try: Divide using synthetic division – Find zeros:

4

x3  8 x 2  11x  20   x  4 
1  8 11 20
4  16  20
1 4  5 0
x  4 x 2  4 x  5
x  4x  5x  1
x4
x5
x  1
You Try: Divide using synthetic
division:
1

 6 x  5x  16 x  10 x  8   x  2 
4
1
2
3
2
6 5  16  10 8
3 4 6 8
6 8  12  16
0
1 3

2
 x   6 x  8 x  12 x  16 
2

You Try: Divide using synthetic
division:

2

x 4  12 x 2  32   x  2 
1 0  12 0
32
 2 4 16  32
1
2
x  2x
3
8
16
0
 2 x  8 x  16
2
