5.4 Dividing Polynomials and Synthetic Division

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What You Will Learn

Divide polynomials by monomials and write

in simplest form.

Use long division to divide polynomials by

polynomials.

Use synthetic division to divide and factor

polynomials.

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Dividing a Polynomial by a Monomial

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Dividing a Polynomial by a Monomial

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Example 1 – Dividing a Polynomial by a Monomial

Perform the division and simplify.

Solution

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cont’d

Example 1 – Dividing a Polynomial by a Monomial

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Long Division

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Example 2 – Long Division Algorithm for Positive Integers

Use the long division algorithm to divide 6584 by 28.

Solution

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So, you have

In Example 2, 6584 is the dividend, 28 is the divisor, 235

is the quotient, and 4 is the remainder.

cont’d

Example 2 – Long Division Algorithm for Positive Integers

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Long Division

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Example 3 – Long Division Algorithm for Polynomials

The remainder is a fractional part of the divisor, so you can

write

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Synthetic Division

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Synthetic Division

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Example 7 – Using Synthetic Division

Use synthetic division to divide x3 + 3x2 – 4x – 10 by x – 2.

Solution

The coefficients of the dividend form the top row of the

synthetic division array. Because you are dividing by x – 2,

write 2 at the top left of the array.

To begin the algorithm, bring down the first coefficient.

Then multiply this coefficient by 2, write the result in the

second row, and add the two numbers in the second

column.

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By continuing this pattern, you obtain the following.

cont’d

Example 7 – Using Synthetic Division

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The bottom row shows the coefficients of the quotient.

So, the quotient is

and the remainder is 2.

So, the result of the division problem is

cont’d

Example 7 – Using Synthetic Division

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Synthetic Division

Synthetic division (or long division) can be used to factor

polynomials.

If the remainder in a synthetic division problem is zero, you

know that the divisor divides evenly into the dividend.

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Example 8 – Factoring a Polynomial

Completely factor the polynomial x3 – 7x + 6 given that one

of its factors is x – 1.

Solution

The polynomial x3 – 7x + 6 can be factored completely

using synthetic division. Because x – 1 is a factor of the

polynomial, you can divide as follows.

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Because the remainder is zero, the divisor divides evenly

into the dividend:

From this result, you can factor the original polynomial as

follows.

x3 – 7x + 6 = (x – 1)(x2 + x – 6)

= (x – 1)(x + 3)(x – 2)

cont’d

Example 8 – Factoring a Polynomial

Homework:

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