Lesson 2.4, page 301
Dividing Polynomials
Objective: To divide polynomials
using long and synthetic division,
and to use the remainder and
factor theorems.
How do you divide a polynomial
by another polynomial?
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Perform long division, as you do with numbers!
Remember, division is repeated subtraction, so
each time you have a new term, you must
SUBTRACT it from the previous term.
Work from left to right, starting with the highest
degree term.
Just as with numbers, there may be a remainder
left. The divisor may not go into the dividend
evenly.
Dividing a Poly by a Binomial
24 33482
See Example 1, page 302.
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Check Point 1:
(x2 + 14x + 45)  (x + 9)
Check Point 2
(7 – 11x - 3x2 + 2x3)  (x - 3)
Missing Terms?
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Write the polynomial in standard form.
If any power is missing, use a zero to
hold the place of that term.
Divide as before.
Check Point 3
(2x4 + 3x3 – 7x - 10)  (x2 – 2x)
Synthetic Division
a simpler process (than long division)
for dividing a polynomial by a binomial;
uses coefficients and part of the divisor
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See Example 4, page 306.
STEPS for Synthetic Division, pg. 306
1) Write polynomial in descending order of the degrees.
2) List the coefficients. (If one power is missing, put a zero
to hold that place.)
3) Write the constant c of the divisor x - c to the left.
4) Bring down the first coefficient.
5) Multiply the first coefficient by c, write the product under
the 2nd coefficient and add.
6) Multiply this sum by c, write it under the next coefficient
and add. Repeat until all coefficients have been used.
7) The numbers on the bottom row are the coefficients of
the answer. The first power on the variable will be one
less than the highest power in the original polynomial.
Check Point 4, page 307
Use Synthetic Division: x3 – 7x – 6 by x + 2.
Caution: What is missing?
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The Remainder Theorem, pg. 307
If the polynomial f(x) is divided by
x – c, then the remainder is the same
value as f(c).
Also:
f(x) = (x – c) q(x) + r
divisor (quotient) + remainder
See Example 5, pg. 308
Check Point 5:
Given f(x) = 3x3 + 4x2 – 5x + 3, use the
remainder theorem to find f(- 4).
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Dividing a Poly by a Binomial
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If a binomial divides into a polynomial with
no remainder, the binomial is a factor of
the polynomial.
Factor Theorem, pg. 308
For the polynomial f(x), if f(c) = 0,
then x – c is a factor of f(x)
Remember . . . If something is a
factor, then it divides the term
evenly with 0 remainder.
See Example 6, pg. 309.
Check Point 6: Solve the equation
15x3 + 14x2 – 3x – 2 = 0, given that -1
is a zero of f(x) = 15x3 + 14x2 – 3x – 2.
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Determine if -1 is a zero of
g(x) = x4 - 6x3 + x2 + 24x -20.