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4.2 Dividing Polynomials
1
Terms you should know:
Quotient
Divisor Dividend
Example using long division:
25
21 540
42
120
105
15 The remainder is 15, the quotient is 25
2x 3 − 2x 2 − 4 x − 6
x−4
Example of long division of polynomials:
2x 2 + 6x + 20 +
x−4
74
x−4
2x 3 − 2x 2 − 4x − 6
− ( 2 x 3 − 8x 2 )
6x 2 − 4x
− (6x 2 − 24x )
20x − 6
− ( 20x − 80)
74
This is your remainder.
Recall: Dividend = Quotient ∗ Divisor + Reminder
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Example 1:
a.
4.2 Dividing Polynomials
Use long division to find the quotient and remainder if any.
2x 4 − 8x 2 + 3x − 5
x−5
b. Use Synthetic Division on the same problem.
2x 4 − 8x 2 + 3x − 5
x−5
2
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4.2 Dividing Polynomials
Example 2:
Divide the following:
6x 3 + 9x 2 + 22x + 2
3x 2 − 2
Example 3:
Divide the following:
x 3 + 3x 2 − 18x − 40
x−4
3
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Example 4:
4.2 Dividing Polynomials
4
Divide the following:
3x 3 + 4x 2 − 13
2x − 4
Here are two theorems that can be helpful in working with polynomials.
The Remainder Theorem: If P(x) is divided by x-c, then the remainder is P(c).
The Factor Theorem: c is a zero of a P(x) if and only if x-c is a factor of P(x), that is if the
remainder when dividing by x-c is zero.
You can use synthetic division and the remainder theorem to evaluate a function at a given value.
Example 5:
Use synthetic division and the remainder theorem to find P(3) for
P( x ) = 2 x 3 − x 2 + 4x + 3
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Example 6:
4.2 Dividing Polynomials
Use synthetic division and the remainder theorem to find P(-1) for
5
P( x ) = 5x 4 − 8x − 5
Example 7: Determine if x + 2 is a factor of P( x ) = x 3 + 6x 2 + 3x − 10 .
Example 8: Show that x = -1 is a zero of P( x ) = 3x 3 − 15x 2 − 3x + 15 . Find the remaining
zeros of the function.
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4.2 Dividing Polynomials
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Example 9: Show that x =2 and x = -3 are zeros of P( x ) = x + 6x + 3x − 26x − 24 .
Find the remaining zeros of the function.
4
3
2
Example 10: Find a 3rd degree polynomial with integer coefficients given that 0, 2 and - 3 are
zeros.