Polynomials
Copyright © Cengage Learning. All rights reserved.
4
Section
4.8
Dividing Polynomials by
Polynomials
Copyright © Cengage Learning. All rights reserved.
Objectives
1 Divide a polynomial by a binomial.
2 Divide a polynomial by a binomial by first
writing exponents in descending order.
3
Divide a polynomial with one or more missing
terms by a binomial.
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1.
Divide a polynomial by a
binomial
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Divide a polynomial by a binomial
To divide one polynomial by another, we use a method
similar to long division in arithmetic.
Recall that the parts of a division problem are defined as
Recall that division by zero is undefined. Therefore, the
divisor cannot be 0. We must exclude any value of the
variable that will result in a divisor of zero.
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Example 1
Divide (x2 + 5x + 6) by (x + 2) Assume no division by 0.
Solution:
Here the divisor is x + 2 and the dividend is x2 + 5x + 6.
We proceed as follows:
Step1:
Step 2:
How many times does x
divide x2?
Write x above the division symbol.
Multiply each item in the divisor by x.
Write the product under x2 + 5x and
draw a line.
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Example 1 – Solution
Step 3:
cont’d
Subtract (x2 + 2x) from (x2 + 5x)
by adding the negative of
(x2 + 2x) to (x2 + 5x).
Bring down the 6.
Step 4:
How many times does x
divide 3x?
= +3
Write +3 above the division symbol.
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Example 1 – Solution
Step 5:
Step 6:
cont’d
Multiply each term in the divisor
by 3. Write the product under the
3x + 6 and draw a line.
Subtract (3x + 6) from (3x + 6)
by adding the negative of
(3x + 6).
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Example 1 – Solution
cont’d
The quotient is x + 3, and the remainder is 0.
Step 7: Check by verifying that x + 2 times x + 3 is
x2 + 5x + 6.
(x + 2)(x + 3) = x2 + 3x + 2x + 6
= x2 + 5x + 6
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2.
Divide a polynomial by a binomial by first
writing exponents in descending order
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Divide a polynomial by a binomial by first writing
exponents in descending order
The division method works best when exponents of the
terms in the divisor and the dividend are written in
descending order.
This means that the term involving the highest power of x
appears first, the term involving the second-highest power
of x appears second, and so on. For example, the terms in
3x3 + 2x2 – 7x + 5
5 = 5x0
have their exponents written in descending order.
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Divide a polynomial by a binomial by first writing
exponents in descending order
If the powers in the dividend or divisor are not in
descending order, we can use the commutative property of
addition to write them that way.
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Example 4
Divide:
Assume no division by 0.
Solution:
We write the dividend so that the exponents are in
descending order and divide.
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Example 4 – Solution
cont’d
Check:
(x + 3)(2x2 – 2x + 4) = 2x3 – 2x2 + 4x + 6x2 – 6x + 12
= 2x3 + 4x2 – 2x + 12
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3.
Divide a polynomial with one or more
missing terms by a binomial
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Divide a polynomial with one or more missing
terms by a binomial
When we write the terms of a dividend in descending
powers of x, we may notice that some powers of x are
missing. For example, if the dividend of
3x4 – 7x2 – 3x + 15,
the term involving x3 is missing. When this happens, we
should either write the term with a coefficient of 0 or leave a
blank space for it.
In this case, we would write the dividend as
3x4 + 0x3 – 7x2 – 3x + 15
or
3x4
– 7x2 – 3x + 15
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Example
Divide:
Assume no division by 0.
Solution:
Since x2 – 4 does not have a term involving x, we must
either include the term 0x or leave a space for it.
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Example – Solution
cont’d
Check:
(x + 2)(x – 2) = x2 – 2x + 2x – 4
= x2 – 4
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