Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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Warm Up: Solve & Sketch the graph: 3 2 () 2 fx x x x 1 2 0 2 0 2 x x x x x x 0 x 2 x 1 x

Transcript of Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Page 1: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Warm Up: Solve & Sketch the graph:

3 2( ) 2f x x x x

120

20 2

xxx

xxx

0x2x

1x

Page 2: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Graphing Polynomials & Finding a Polynomial

Function

Page 3: Warm Up: Solve & Sketch the graph:. 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

Page 4: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Multiplicity

Page 5: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Repeated Zeros

.

,1,,

kymultiplictofaxzerorepeatedayields

kaxoffactorafunctionpolynomialaFor k

.

,.2

.

,.1

axat

axisxtheTOUCHESgraphtheEVENiskIf

axat

axisxtheCROSSESgraphtheODDiskIf

Page 6: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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:

Page 7: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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.

Page 8: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 9: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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:

Page 10: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Can you write a polynomial function if you know the zeros?

Page 11: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Find a Polynomial function that has the given zeros: 6, -3

1863

36

36

2

xxx

xx

xx

1832 xx

Page 12: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 13: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 14: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Find a Polynomial function that has the given zeros: and 5 2 5 2

22522525525

2525

2525

2

xxxxx

xx

xx

23102 xx

Page 15: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Long & Synthetic Division

Page 16: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

3 22 6 19 16 4x x x x

Use Long Division and use the resultto factor the polynomial completely.

1st Step:

Page 17: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

2nd Step:

23122

2762 2

xxx

xxx

Page 18: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 19: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 20: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 21: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 22: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Synthetic Division

Page 23: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.
Page 24: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 25: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Divide using synthetic division:

4 210 2 4 3x x x x

3 421001

1131

3393

1

3

1133 23

xxxxx

Page 26: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 27: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

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

Page 28: Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

You Try: Divide using synthetic division:

4 212 32 2x x x

2

16822 23 xxxx

321642

3201201

016821