Dividing polynomials

This PowerPoint presentation demonstrates two different methods

of polynomial division.

© Kristen A. Treglia 2007

Algebraic long division

Divide 2x³ + 3x² - x + 1 by x + 2

3 22 2 3 1x x x x x + 2 is the divisor

The quotient will be here.

2x³ + 3x² - x + 1 is the dividend

Algebraic long division

First divide the first term of the dividend, 2x³, by x (the first term of the divisor).

3 22 2 3 1x x x x

22xThis gives 2x². This will be the first term of the quotient.

Algebraic long division

Now multiply 2x² by x + 2

3 22 2 3 1x x x x 3 22 4x x

22x

2xand subtract

Algebraic long division

Bring down the next term, -x.

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

Algebraic long division

Now divide –x², the first term of –x² - x, by x, the first term of the divisor

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

which gives –x.

Algebraic long divisionMultiply –x by x + 2

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x To get a result of:

Algebraic long division

Now subtract

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x

** remember, you must subtract both the –x2 term and the -2x term

Algebraic long division

Now combine like terms

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x

x

(since we subtracted two negative terms they both turned into positive terms)

{

Algebraic long division

Bring down the next term, 1 x

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x

1

Algebraic long division

Divide x, the first term of x + 1, by x, the first term of the divisor

13 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x

x 1which gives 1

Algebraic long division

Multiply x + 2 by 1

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x

x

1

12x 1and subtract

Algebraic long division

The remainder is –1.

3 22 2 3 1x x x x 3 22 4x x

22x

2x x

x

2 2x x

x

1

12x 1

The quotient is 2x² - x + 1