warm up: solve & sketch the graph:. graphing polynomials & finding a polynomial function
TRANSCRIPT
Warm Up: Solve & Sketch the graph:
3 2( ) 2f x x x x
120
20 2
xxx
xxx
0x2x
1x
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
.
,1,,
kymultiplictofaxzerorepeatedayields
kaxoffactorafunctionpolynomialaFor k
.
,.2
.
,.1
axat
axisxtheTOUCHESgraphtheEVENiskIf
axat
axisxtheCROSSESgraphtheODDiskIf
Sketch the graph: 4 2( ) 2 2f x x x
1120
120
220
2
22
24
xxx
xx
xx
110 xxx
Multiplicity of 2.Touches.
Through thesePoints.
End behavior:
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: 3 2( ) 3 ( 2)f x x x
20
230 23
xx
xx
Mult. of 3Goes Through
Mult. Of 2Touches
1st Term would be .3 5xEnd Behavior:
1 3
x y
Sketch the graph: 2 2( ) ( 2)f x x x
20
20 22
xx
xx
Both have a multiplicityof 2.Just Touch!
x y-1 -1
1st Term would be .4xEnd Behavior:
Can you write a polynomial function if you know the zeros?
Find a Polynomial function that has the given zeros: 6, -3
1863
36
36
2
xxx
xx
xx
1832 xx
Find a Polynomial function that has the given zeros: 0, 3, -5
xxxx
xxx
xxx
xxx
1535
53
53
530
223
2
xxx 152 23
Find a Polynomial function that has the given zeros: 3, 2, -2, -1
121861015523
2365
1223
1223
223234
22
xxxxxxxx
xxxx
xxxx
xxxx
12872 234 xxxx
Find a Polynomial function that has the given zeros: and 5 2 5 2
22522525525
2525
2525
2
xxxxx
xx
xx
23102 xx
Long & Synthetic Division
3 22 6 19 16 4x x x x
Use Long Division and use the resultto factor the polynomial completely.
1st Step:
2nd Step:
23122
2762 2
xxx
xxx
Divide using long division:
3 23 2 17 17 3x x x x
1717233 23 xxxx 3
5473 2
xxx
23 93 xx
xx
xx
217
1772
2
124
174
x
x
5Remainder
Divide using long division:
3 64 4x x
64004 23 xxxx
1642 xx
23 4xx
xx
xx
164
042
2
6416
6416
x
x
0
1644 2 xxx
You Try: Divide using long division
3 26 20 10 5 2x x x x
5102062 23 xxxx
2686 2 xx
23 126 xx
xx
xx
168
1082
2
5226
526
x
x
47: R
2
47
x
Divide using long division
4 3 2 22 4 5 3 2 2 3x x x x x x
2354232 2342 xxxxxx
32
112
22
xx
xx
Synthetic Division
Divide using synthetic division to find the zeros:
2 10 21 3x x x
3 21101
1
3
7
21
0
73
073
xx
xx
Divide using synthetic division:
4 210 2 4 3x x x x
3 421001
1131
3393
1
3
1133 23
xxxxx
You Try: Divide using synthetic division – Find zeros:
3 28 11 20 4x x x x
4
154
544 2
xxx
xxx154 xxx
201181
1
4
416
5
20
0
You Try: Divide using synthetic division:
4 3 2 16 5 16 10 8
2x x x x x
2
1
1612862
1 23
xxxx
8643
8101656
0161286
You Try: Divide using synthetic division:
4 212 32 2x x x
2
16822 23 xxxx
321642
3201201
016821