Algebra 1 > Notes > Yorkcounty Final > Yc > Unit 8 - Polynomial > Polynomial Division

Polynomial Division

Recall from arithmetic that

2 + 10 + 8 2

is the same as

2 10 8 + + 2 2 2 This is a skill which we can apply to expressions like the one you saw in the title slide of this presentation. There was a polynomial 2x2+10x2+8x divided by a monomial of 2x. We can use the property from arithmetic above to make it into three separate expressions: 3

2

2 x 10 x 8x + + 2x 2x 2x

3

2

2 x 10 x 8x + + 2x 2x 2x For each of the three expressions we have two things to divide: numbers and variables. Let’s do the numbers first…… 1

5 3

4 2

2 x 10 x 8x + + 2x 2x 2x 1

1

1

This leaves us with:

3

2

1x 5x 4x + + 1x 1x 1x To simplify the variables we will need another exponent property called the “Quotient of Powers Property” Let’s look at an example to help us understand this property: 3

x 2 x

Now lets use what we know about exponents to expand the situation. 3

x x⋅x⋅x = 2 x x⋅x

Now since we know x/x = 1 we can cross out the common x’s

x⋅ x⋅ x x⋅ x

This leaves us with one x1:

=x

1

The answer is x1 or x.

We could have just subtracted the exponents of x like this: 3

x 3− 2 1 = x = x =x 2 x

This brings us to the Quotient m

Powers Property:

a m− n =a n a When taking the quotient of powers with the same base, simply subtract the exponents.

Now lets return to our problem: We started with…. 2 x 3 10 x 2 8 x + + 2x 2x 2x And then simplified it to…… 2 x 3 10 x 2 8 x + + 2x 2x 2x

And then divided the coefficients to get…

1x 3 5x 2 4 x 1 + 1+ 1 1 1x 1x 1x

Don’t forget that all the variables which have no exponents really have exponents of 1. This makes it look like...

1x 2 + 5x + 4 If we subtract our exponents in each part we get….. 3−1 2 −1 1−1

1x

+ 5x

+ 4x

Which simplifies to…. 2 1

1x + 5x + 4 x

0

Recall that raising to a zero power means 1….

1x 2 + 5x + 4(1)

And finally we get:

1x 2 + 5x + 4

Try This  Divide

x3 by x

3

x 3−1 2 =x =x x

Practice  Divide

6x5 by 3x3 2

5

6x 5−3 2 = 2 x = 2 x 3 3x 1

Try This 

Divide 28a9 ÷ 4a3

Solution on the next page

Solution Divide

9

28a9 ÷ 4a3

9

28a 7a 9−3 6 = = 7 a = 7 a 3 3 4a 1a

Try This 

Divide: (20n4-15n3+35n2)÷5n2



solution on next slide

Solution:

20n − 15n + 35n 5n 2 4

3

2

20n 4 15n 3 35n 2 = − 2 + 2 2 5n 5n 5n

4n 4 3n 3 7n 2 = 2 − 2+ 2 1n 1n 1n

= 4n

4−2

− 3n

3− 2

= 4n − 3n + 7n 2

1

= 4n 2 − 3n1 + 7(1)

= 4n − 3n + 7 2

0

+ 7n

2−2

Last One…

32n − 24n + 40n 3 8n 5

4

3

SOLUTION:

32n5 − 24n 4 + 40n 3 8n 3 32n5 24n 4 40n 3 = − + 3 3 8n 8n 8n 3

= 4n

5− 3

− 3n

4−3

+ 5n

= 4n − 3n + 5n 2

1

= 4n 2 − 3n1 + 5(1)

= 4n 2 − 3n + 5

0

3− 3