1. Digital Lesson
Polynomial Functions

2. A polynomial function is a function of the form
where n is a nonnegative integer and each ai (i = 0, , n) is a real number.
The polynomial function has a leading
coefficient an and degree n.
Examples: Find the leading coefficient and degree of each polynomial function.
Polynomial Function Leading Coefficient Degree –
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Polynomial Function

3. A polynomial function of degree n has at most n zeros.
A real number a is a zero of a function y = f (x) if and only if f (a) = 0.
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).
A polynomial function of degree n has at most n zeros.
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Zeros of a Function

4. Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2.
Factor completely:
f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2). y x –2 2
f (x) = x4 – x3 – 2x2
The real zeros are x = –1, x = 0, and x = 2.
(–1, 0)
(0, 0)
These correspond to the x-intercepts.
(2, 0)
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Example: Real Zeros

5. Dividing Polynomials
Example: Divide x2 + 3x – 2 by x – 1 and check the answer. 1. x
+ 2 2.
x x 3. 2x
– 2
2x + 2 4.
– 4 5.
remainder 6.
Answer: x + 2 +
– 4
Check:
(x + 2)
quotient
(x + 1)
divisor
+ (– 4)
remainder
= x2 + 3x – 2
dividend
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Dividing Polynomials

6. Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division.
Synthetic division is a shorter method of dividing polynomials. This method can be used only when the divisor is of the form x – a. It uses the coefficients of each term in the dividend.
Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division.
Since the divisor is x – 2, a = 2.
value of a
coefficients of the dividend
1. Bring down 3 2
– 1
2. (2 • 3) = 6
3. (2 + 6) = 8 6 16
4. (2 • 8) = 16 3 8 15
5. (–1 + 16) = 15
coefficients of quotient
remainder
3x + 8
Answer: 15
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Synthetic Division

7. The remainder is 68 at x = 3, so f (3) = 68.
Remainder Theorem: The remainder of the division of a polynomial f (x) by x – k is f (k).
Example: Using the remainder theorem, evaluate f(x) = x 4 – 4x – 1 when x = 3.
value of x 3
– – 1 3 9 27 69 1 3 9 23 68
The remainder is 68 at x = 3, so f (3) = 68.
You can check this using substitution:
f(3) = (3)4 – 4(3) – 1 = 68.
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Remainder Theorem

8. The remainders of 0 indicate that (x + 2) and (x – 1) are factors.
Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k) = 0.
Example: Show that (x + 2) and (x – 1) are factors of f(x) = 2x 3 + x2 – 5x + 2.
– 2 – 1 –
– 4 6
– 2 2
– 1 2
– 3 1 2
– 1
The remainders of 0 indicate that (x + 2) and (x – 1) are factors.
The complete factorization of f is (x + 2)(x – 1)(2x – 1).
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Factor Theorem

9. where p and q have no common factors other than 1.
Rational Zero Test: If a polynomial f(x) has integer coefficients, every rational zero of f has the form
where p and q have no common factors other than 1.
p is a factor of the constant term.
q is a factor of the leading coefficient.
Example: Find the rational zeros of f(x) = x3 + 3x2 – x – 3.
q = 1
p = – 3
The possible rational zeros are ±1 and ±3.
Synthetic division shows that the factors of f are (x + 3), (x + 1), and (x – 1).
The zeros of f are – 3, – 1, and 1.
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Rational Zero Test

10. Descartes’s Rule of Signs
Descartes’s Rule of Signs: If f(x) is a polynomial with real coefficients and a nonzero constant term,
The number of positive real zeros of f is equal to the number of variations in sign of f(x) or less than that number by an even integer.
The number of negative real zeros of f is equal to the number of variations in sign of f(–x) or less than that number by an even integer.
A variation in sign means that two consecutive, nonzero coefficients have opposite signs.
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Descartes’s Rule of Signs

11. Example: Descartes’s Rule of Signs
Example: Use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros of f(x) = 2x4 – 17x3 + 35x2 + 9x – 45.
The polynomial has three variations in sign.
+ to –
+ to –
f(x) = 2x4 – 17x3 + 35x2 + 9x – 45
– to +
f(x) has either three positive real zeros or one positive real zero.
f(– x) = 2(– x)4 – 17(– x)3 + 35(– x)2 + 9(– x) – 45
=2x4 + 17x3 + 35x2 – 9x – 45
One change in sign
f(x) has one negative real zero.
f(x) = 2x4 – 17x3 + 35x2 + 9x – 45 = (x + 1)(2x – 3)(x – 3)(x – 5).
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Example: Descartes’s Rule of Signs

12. Use your graphing calculator and division to find out the zero that is not an integer for the function f(x)=2x4 – 17x3 + 35x2 + 9x – 45.
2x4 – 17x3 + 35x2 + 9x – 45
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13. Graphing Utility: Finding Roots
Graphing Utility: Find the zeros of f(x) = 2x3 + x2 – 5x + 2.
– 10 10
Calc Menu:
The zeros of f(x) are x = – 2, x = 0.5, and x = 1.
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Graphing Utility: Finding Roots

14. Homework: p #1,2,8,13,14,31,35,39-41,49
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15. 1. Divide using long division.
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16. 1. Divide using synthetic division.
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18. 8.Divide using long division
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23. Use synthetic division to find each function value
Use synthetic division to find each function value. Use a graphing utility to verify. 31. a. f(1) b. f(-2) c. f(1/2) d. f(8)
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24. Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function.
x=2
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29. Use the graphing calculator to find the one exact zero
Use the graphing calculator to find the one exact zero. Then use synthetic division to break down the function to find the other two exact zeros. 53.
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30. Use Descartes’s Rule of Signs to determine the possible number of + and – real zeros. Use a graphing calculator to verify. 61.
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