2.2 polynomial function notes
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Warm-Up• Sketch the graphs of the following:
f(x)=x f(x)=x2 f(x)=x3 f(x)=x4 f(x)=x5
End Behavior:Even functions either start up and end up or start down and end down
Odd functions either start down and end up or start up and end down.
Match the equations with their graph.
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21( ) 3 5
2 f x x x
5 4 3( ) 2 2 5 2 f x x x x x
3 2( ) 3 6 f x x x x
4 2( ) 3 5 f x x x
2( ) 4 f x x x
( ) 5 f x x
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A polynomial function is a function of the form1
1 1 0( ) , 0n nn n nf x a x a x a x a a
where n is a nonnegative integer and each ai is a real number.
The polynomial function has a leading coefficient an and degree n.
Examples:5 3( ) 2 3 5 1f x x x x
3 2( ) 6 7f x x x x ( ) 14f x
Section 2.2
Solve the following
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20 3 2 x x
20 3 2
0 ( 1)( 2)
1 0 2 0
1 2
x x
x x
x x
x x
There are multiple ways to write the answers.
x=1 is a zero
x=1 is a solution
x-1 is a factor
(1,0) is an x-intercept
The correct ways depends on the question.
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A real number a is a zero of a function y = f (x)if and only if f (a) = 0.
A polynomial function of degree n has at most n zeros.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.
1. x = a is a zero of f.
2. x = a is a solution of the polynomial equation f (x) = 0.
3. (x – a) is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
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y
x–2
2
Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2.
Factor completely: f (x) = x 4 – x3 – 2x2
The real zeros are x = -1,x=0 double root, and x = 2.
When the roots are real the zeros correspond to the x-intercepts. f (x) = x4 – x3 – 2x2
(–1, 0) (0, 0)
(2, 0)
= x2(x2 – x – 2)
= x2(x + 1)(x – 2)
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Graphing Utility: Find the zeros of f(x) = 2x3 + x2 – 5x + 2.
Calc Menu:
The zeros of f(x) are x = – 2, x = 0.5, and x = 1.
– 10 10
10
– 10
Solve for the zeros using a graphing calculator.
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3 21. 3 4 15 20 y x x x
5 22. 13 5 y x x
3 23. 3 8 y x x x
3 24. 8 12 y x x x
Write the polynomial with the following roots.
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1. 3, 2 x
2. 3,0x
3. 2 5, 4 x
4. 3 , 2,0 x double root