This PowerPoint presentation demonstrates two methods of polynomial division: algebraic long division and synthetic division. It provides a step-by-step example of using algebraic long division to divide the polynomial 2x^3 + 3x^2 - x + 1 by the polynomial x + 2. The quotient is 2x^2 - x + 1 with a remainder of -1.

1. Dividing polynomials
This PowerPoint presentation
demonstrates two different methods
of polynomial division.

2. Algebraic long division
Divide 2x³ + 3x² - x + 1 by x + 2
3 2
2 2 3 1
x x x x
   
x + 2 is the
divisor
The quotient
will be here.
2x³ + 3x² - x + 1
is the dividend

3. Algebraic long division
First divide the first term of the dividend, 2x³,
by x (the first term of the divisor).
3 2
2 2 3 1
x x x x
   
2
2x
This gives 2x².
This will be the
first term of
the quotient.

4. Algebraic long division
Now multiply
2x² by x + 2
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x

and subtract

5. Algebraic long division
Bring down the
next term, -x.
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x


6. Algebraic long division
Now divide –x²,
the first term of
–x² - x, by x, the
first term of the
divisor
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

which gives –x.

7. Algebraic long division
Multiply –x by x + 2
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
To get a result of:

8. Algebraic long division
Now subtract
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
** remember, you must
subtract both the –x2
term and the -2x term

9. Algebraic long division
Now combine
like terms
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
x
(since we subtracted two
negative terms they both
turned into positive terms)
{

10. Algebraic long division
Bring down the
next term, 1
x
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
1


11. Algebraic long division
Divide x, the first
term of x + 1, by x,
the first term of
the divisor
1

3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
x 1

which gives 1

12. Algebraic long division
Multiply x + 2 by 1
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
x
1

1

2
x 
1

and subtract

13. Algebraic long division
The remainder is –1.
3 2
2 2 3 1
x x x x
   
3 2
2 4
x x

2
2x
2
x
 x

x

2
2
x x
 
x
1

1

2
x 
1

The quotient is
2x² - x + 1