Lesson 2.3 Real Zeros of Polynomials

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Lesson 2.3Real Zeros of Polynomials
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The Division Algorithm
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Dividing by a polynomial Set up in long
division
2 terms in divisor (x 1). How does this go into
1st two terms in order to eliminate the 1st term
of the dividend.
2x
1

• Multiply by the divisor
• Write product under dividend
• Subtract
• Carry down next term
• Repeat process

2x2 2x
-
-
x 5
x 1
-
-
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Answer
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HINTS If a term is missing in the dividend add
a 0 term. If there is a remainder, put it over
the divisor and add it to the quotient (answer)
Example 1 (x4 x2 x) (x2 - x 1)
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• Synthetic Division
• Less writing
• Uses addition

• Setting Up
• Divisor must be of the form x a
• Use only a and coefficients of dividend
• Write in zero terms

x 2 a 2 x 3 a -3
4 5 0 -2 5
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4 5 0 -2 5
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• Steps
• Bring down
• Multiply diagonally
• Add
• Remainder last addition
• Answer
• Numbers at bottom are coefficients
• Start with 1 degree less than dividend

REPEAT
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Example 2 (2x3 7x2 11x 20) (x 5)

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Example 3 (2x4 30x2 2x 1) (x 4)

Problem Set 2.3 (1 21 EOO)
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The Remainder Theorem If f(x) is divided by x
a , the remainder is r f(a)
The Factor Theorem If f(x) has a factor (x a)
then f(a) 0
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Example 4 Show that (x 2) and (x 3) are
factors of
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Rational Zero Test Every rational zero
Factors of constant term
Factors of leading coefficient

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Descartes Rule Number of positive real roots
is ? the number of variations in the signs,
or ? less than that by a positive even
integer 5x4 3x3 2x2 7x
1 variations possible positive real roots
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Example 5 List possible zeros, verify with your
calculator which are zeros, and check results
with Descartes Rule
Problems Set 2.3