2.3 real zeros of polynomial functions 2015 digital lesson
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2.3Real Zeros of
Polynomial Functions2015
Digital Lesson
Warm-up
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4 24f x x x
Find the zeros of the function andSketch a graph with those zeros and correct end
behavior.
Objective
• To find real zeros of polynomial functions using various methods.
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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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+ 2 2 3 1 2 xxx
Example: Divide x2 + 3x – 2 by x + 1 and check the answer.
x
x2 + x
2x – 22x + 2
– 4
remainder
Check: (x + 2)
quotient
(x + 1)
divisor
+ (– 4)
remainder
= x2 + 3x – 2
dividend
Answer: x + 2 +1x
– 4
Dividing Polynomials
You try:
Divide by using long division.
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22 10 12x x 3x
Try another one:
Divide using long division.
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( ) ( )x x x x4 2 23 1 2 3
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16
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.
3 2 – 12
Since the divisor is x – 2, a = 2.
3
6
8 15coefficients of quotient remainder
value of a coefficients of the dividend
3x + 8Answer: 2
x15
1. Divide using synthetic division.
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2 10 12 32x x x
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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.
6
2 1 – 5 2– 2
2
– 4
– 3 1
– 2
0
The remainders of 0 indicate that (x + 2) and (x – 1) are factors.
– 1
2 – 3 11
2
2
– 1 0
The complete factorization of f is (x + 2)(x – 1)(2x – 1).
Find all zeros of the function given x+2 is a factor.
• f(x) = 2x3 + x2 – 5x + 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
Use the graphing calculator to find the exact value of all real zeros.
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f x x x x( ) 3 22 7 2
Verify that (x+2) and (x-4) are factors of the function and find any remaining factors to write the complete factorization of f. List all real zeros.
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4 3 2( ) 8 14 71 10 24f x x x x x
• Homework:
• Pg. 123 1-33 EOO, 35-43 odd ,49-55 odd
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HWQ practice
Solve by dividing out a zero and factoring the resulting quadratic ( or use quadratic formula):
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3 26 11 24 9 0x x x
Use the graphing calculator to find the any exact zeros. Then use synthetic division to break down the function to find the other zeros.
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4 3 2( ) 5 3f x x x x x
Find all real solutions of the polynomial. Use the rational root test, or graphing calculator. Extra credit to anyone who can factor the polynomial by hand.
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5 4 3 23 5 2 0x x x x x
• Homework:
2.1/2.3 Review Worksheet
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2.3Real Zeros of
Polynomial FunctionsDay 2 – More
problems
Digital Lesson
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