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Dividing Polynomials

Long Division of Polynomials

- Arrange the terms of both the dividend and the

divisor in descending powers of any variable. - Divide the first term in the dividend by the

first term in the divisor. The result is the

first term of the quotient. - Multiply every term in the divisor by the first

term in the quotient. Write the resulting product

beneath the dividend with like terms lined up. - Subtract the product from the dividend.
- Bring down the next term in the original dividend

and write it next to the remainder to form a new

dividend. - Use this new expression as the dividend and

repeat this process until the remainder can no

longer be divided. This will occur when the

degree of the remainder (the highest exponent on

a variable in the remainder) is less than the

degree of the divisor.

Text Example

Divide 4 5x x2 6x3 by 3x 2.

Text Example cont.

Divide 4 5x x2 6x3 by 3x 2.

The Division Algorithm

If f (x) and d(x) are polynomials, with d(x) 0,

and the degree of d(x) is less than or equal to

the degree of f (x), then there exist unique

polynomials q(x) and r(x) such that f (x)

d(x) q(x) r(x). The remainder, r(x),

equals 0 or its is of degree less than the degree

of d(x). If r(x) 0, we say that d(x) divides

evenly in to f (x) and that d(x) and q(x) are

factors of f (x).

Synthetic Division

To divide a polynomial by x c Example 1.

Arrange polynomials in descending powers,

with a 0 coefficient for any missing terms. x

3 x3 4x2 5x 5 2. Write c for the

divisor, x c. To the right, 3 1 4 -5

5 write the coefficients of the dividend. 3.

Write the leading coefficient of the dividend 3

1 4 -5 5 on the bottom row. Bring

down 1. 1 4. Multiply c (in this case,

3) times the value 3 1 4 -5 5 just

written on the bottom row. Write the 3

product in the next column in the 2nd row. 1

Synthetic Division

5. Add the values in this new column, writing

the sum in the bottom row.

6. Repeat this series of multiplications and

additions until all columns are filled

in. 7. Use the numbers in the last row to

write the quotient and remainder in

fractional form. The degree of the first

term of the quotient is one less than the

degree of the first term of the dividend.

The final value in the row is the remainder.

Text Example

Use synthetic division to divide 5x3 6x 8 by

x 2.

Text Example cont.

Solution We begin the synthetic division

process by bringing down 5. This is following by

a series of multiplications and additions.

Text Example cont.

The Remainder Theorem

- If the polynomial f (x) is divided by x c, then

the remainder is f (c).

The Factor Theorem

- Let f (x) be a polynomial.
- If f (c ) 0, then x c is a factor of f (x).
- If x c is a factor of f (x), then f ( c) 0.

Text Example

Solve the equation 2x3 3x2 11x 6 0 given

that 3 is a zero of f (x) 2x3 3x2 11x 6.

Text Example cont.

Solution

Now we can solve the polynomial equation.

The solution set is -2, 1/2 , 3.

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