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. 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