3.2 Polynomial Functions & Models November 10, 2008
3.2 Polynomial Functions & Models
Objectives: • Identify polynomial funtions and their degree
• Find polynomial functions from their zeros
• Identify the zeros of a polynomial function and their multiplicity
Nov 4­2:30 PM
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3.2 Polynomial Functions & Models November 10, 2008
Polynomial Function
f(x) = anxn + an­1xn­1 + ... + a1x + a0
where an, an ­ 1, ... , a1, a0 are real numbers and n is a nonnegative integer.
The domain is the set of all real numbers.
The degree is the largest power of x that appears.
Nov 4­2:34 PM
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3.2 Polynomial Functions & Models November 10, 2008
Objectives: •Identify polynomial funtions and their degree
• Find polynomial functions from their zeros
• Identify the zeros of a polynomial function and their multiplicity
Nov 4­3:16 PM
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3.2 Polynomial Functions & Models November 10, 2008
Determine which of the following are polynomial functions. For those that are, state the degree; for those that are not, tell why not.
1. f(x) = 7x2 + 3x4
Yes, degree 4
2. f(x) = ­3x2(4x ­ 1)
Yes, degree 3
3. f(x) = √x + 8
No, x is raised
to the 1/2 power.
4. f(x) = 4x2(x + 3)3
Yes, degree 5
5. f(x) = x2 + 3
x3 ­ 5
No, the degree in
the denominator is +3
Nov 4­2:41 PM
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3.2 Polynomial Functions & Models November 10, 2008
Objectives: • Identify polynomial funtions and their degree
•Find polynomial functions from their zeros
• Identify the zeros of a polynomial function and their multiplicity
Nov 4­3:16 PM
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3.2 Polynomial Functions & Models November 10, 2008
EXAMPLE 1:
Find a polynomial of degree 3 whose zeros are ­4, 2 and 3
*If ­4 is a zero of a polynomial, then x + 4 is a factor of that polynomial. *If 2 and 3 are zeros, then x ­ 2 and x ­ 3 are also factors.
*Therefore, this polynomial function is of the form
f(x) = a(x + 4)(x ­ 2)(x ­ 3)
where a is any nonzero real number.
*The value of a causes a stretch, compression or reflection but does not affect the x­intercepts so we say...
f(x) = (x + 4)(x ­ 2)(x ­ 3) for a = 1
*Remember to simplify:
f(x) = x3 ­ x2 ­ 14x + 24 for a = 1
Nov 4­2:51 PM
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3.2 Polynomial Functions & Models November 10, 2008
EXAMPLE 2:
Find a polynomial of degree 3 whose zeros are ­3 multiplicity 2 and 5 multiplicity 1
*Multiplicity ­ describes the number of times the factor occurs
f(x) = a(x + 3)2(x ­ 5)
f(x) = x3 + x2 ­ 21x ­ 45
for a = 1
Nov 4­3:12 PM
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3.2 Polynomial Functions & Models November 10, 2008
Objectives: • Identify polynomial funtions and their degree
• Find polynomial functions from their zeros
• Identify the zeros of a polynomial function and their multiplicity
Nov 4­3:16 PM
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3.2 Polynomial Functions & Models November 10, 2008
SDRAWKCAB GNIKROW
Identify the zeros and their multiplicities.
1.
f(x) = 3(x ­ 2)(x + 1)3
2.
3.
f(x) = ­4(x + 1/2)2(x ­ 5)3
2 multiplicity 1
­1 multiplicity 3
­1/2 multiplicity 2
5 multiplicity 3
f(x) = (x ­ √2)2(x ­ 4)4
√2 multiplicity 2
4 multiplicity 4
Nov 4­3:07 PM
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3.2 Polynomial Functions & Models November 10, 2008
Multiplicity:
Even ­ graph touches x­axis at the zero
*Sign of f(x) does not change from one side of the zero to the other side of the zero.
Odd ­ graph crosses x­axis at the zero
*Sign of f(x) changes from one side of the zero to the other side of the zero.
Nov 4­3:58 PM
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3.2 Polynomial Functions & Models November 10, 2008
Determine whether the graph crosses or touches the x­axis at each x­intercept.
1.
f(x) = 3(x ­ 2)(x + 1)3
crosses at ­1 & 2 2.
3.
f(x) = ­4(x + 1/2)2(x ­ 5)3
touches at ­1/2 & crosses at 5
f(x) = (x ­ √2)2(x ­ 4)4
touches at √2 & 4
Nov 4­3:07 PM
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3.2 Polynomial Functions & Models November 10, 2008
Power Function:
The power function of degree n is a function of the form
f(x) = axn
where a is a real number, a ≠ 0, and n > 0 is an integer.
Examples:
f(x) = 3x f(x) = ­5x2
f(x) = 8x3
f(x) = ­5x4
degree 1
degree 2
degree 3
degree 4
Nov 4­4:06 PM
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3.2 Polynomial Functions & Models November 10, 2008
Find the power function that the graph of f resembles for large values of |x|.
1.
f(x) = 3(x ­ 2)(x + 1)3
f(x) = 3x4 2.
3.
f(x) = ­4(x + 1/2)2(x ­ 5)3
f(x) = ­4x5
f(x) = (x ­ √2)2(x ­ 4)4
f(x) = x6
Nov 4­4:04 PM
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3.2 Polynomial Functions & Models November 10, 2008
Homework: page 182 (12 ­ 22 even, 38 ­ 44 even, 46 ­ 56 even)
Nov 4­3:24 PM
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