Division of Polynomials

The Long and Short of it.

Review of Vocabulary

Factorroot, zero, solution, x-interceptdividend ÷ divisor = quotientExample: in 12 ÷ 3 = 4:

12 is the dividend3 is the divisor4 is the quotient

remainder

Polynomial Long Division

• If you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables.

Dividing by a binomial

• The next few slides demonstrate how to do the long division. The slides are meant as background information. You will find the same steps at :

PurplemathI will demonstrate a quicker way also.

Example 1

• Divide x2 – 9x – 10 by x + 1• The process is similar to long

division of numbers as you learned in elementary school.

First, I set up the division: For the moment, I'll ignore the other terms and look just at the leading x of the divisor and the leading x2 of the dividend.

If I divide the leading x2 inside by the leading x in front, what would I get? I'd get an x. So I'll put an x on top: 

 

Now I'll take that x, and multiply it through the divisor, x + 1. First, I multiply the x (on top) by the x (on the "side"), and carry the x2 underneath: 

Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath:   

Then I'll draw the "equals" bar, so I can do the subtraction. To subtract the polynomials, I change all the signs in the second line...   

 

   ...and then I add down. The first term (the x2) will cancel out:    

   I need to remember to carry down that last term, the "subtract ten", from the dividend:   

Now I look at the x from the divisor and the new leading term, the –10x, in the bottom line of the division. If I divide the –10x by the x, I would end up with a –10, so I'll put that on top:

 

 

Now I'll multiply the –10 (on top) by the leading x (on the "side"), and carry the –10x to the bottom:   

   ...and I'll multiply the –10 (on top) by the 1 (on the "side"), and carry the –10 to the bottom:   

   I draw the equals bar, and change the signs on all the terms in the bottom row:   

   

   Then I add down:   

finally

• The answer is x-10.• That means  x2 – 9x – 10 divide by

x + 1= x-10.In another words:x2 – 9x – 10 = (x + 1)(x-10)One more way to say: x-10 is a factor of x2 – 9x – 10

Why is division useful?

When solving a quadratic such as You may factor the quadratic first and then find the roots of each linear factor.

therefore Or therefore

As useful as long division is, most people find the steps difficult to follow.

A quicker way to divide is to use synthetic division (AKA synthetic substitution)

The next few slide will walk you through the steps. Please take careful notes.

Before you start to use Synthetic Division, remember:

• 1. Both dividend and divisor have to be in the standard form.

• 2. If a term is missing in dividend, replace it with a zero

• 3. The leading co-efficient of the divisor has to be 1.

Step 1

• DIVIDEND: Strip off all of the variables and consider only coefficients. Remember to insert zero for each missing term.

• Find the zero of the divisor.

Bring the leading coefficient of the dividend down. Then there is a series of multiplication and addition.

The last term is the remainder.

If remainder (last term) is zero, you have found a factor.

Example 1

OR The dividend is: ; consider only it coefficients:1 16 59 -6The divisor is ; its zero is -6 since

-6 1 16 59 -6

1

-6

10

-60

-1

6

0

x2 + 10x - 1 R=0

Oppositeof number indivisor

Therefore

Therefore in order to find all the zeros of ; one would find the roots of the linear function x+6 and the roots of the quadratic The roots of are:-6, 0.1 and -10.1

Example 2

The dividend is: ; consider only it coefficients:4 18 12 -18The divisor is ; its zero is -3 since

-3 4 18 12 -18

4

-12

6

-18

-6

18

0

4x2 + 6x - 6 R=0

Oppositeof number indivisor

Synthetic division and substitution

• Synthetic division is also referred to as synthetic substitution, because the remainder is value of the polynomial if the zero of the divisor is substituted in for the variable.

Example 3:

-7 1 6 -15 -52

1

-7

-1

7

-8

56

4

R=4

Oppositeof number indivisor

Ex 3 Continued:

• Now subsitute x= -7 into

• • Like magic, answer is 4 which is the

remainder.• Evaluating polynomials becomes

simpler using this method, which usually does not require a calculator.