Long Division of Polynomials

A different way to factor (7.5)

SAT Prep

Quick poll!

1.

SAT Prep

Quick poll!

2.

SAT Prep

Quick poll!

3.

POD

Factor:

x2 – x – 12

Can you use that pattern to factor this one completely?

x4 – x2 – 12

Pattern review

Difference of squares: a2 – b2 = (a + b)(a - b)

Difference of cubes: a3 – b3 = (a - b)(a2 + ab + b2)

Sum of cubes: a3 + b3 = (a + b)(a2 - ab + b2)

Factoring by grouping (split the middle term)

If the discriminant is a perfect square, then the quadratic trinomial can be factored.

Factoring and division

If

then

So, what is ?

What are the remainders with these? What does that mean?

baba

ba

22

22))(( bababa

ba

ba

33

Division with a remainder

We can divide polynomials, even if we get a remainder that isn’t zero. In that case, we don’t factor, but do something called long division of polynomials. It’s a lot like long division of numbers.

Division with a remainder

First, let’s review the following terms, and do a simple long division problem.

dividend

divisorremainder

quotient

3665

Division with a remainder

Here’s how it works for polynomials. becomes

The tricky part to keep straight is what is positive and negative.

A remainder of 0 means we have factors.

2

141323 23

x

xxx 1413232 23 xxxx

Try another

Use the same technique here. 2

1232 23

x

xxx

Try another

This time the divisor is not linear, but the process is the same.

12

1332

23

xx

xxx

Try another

You may find this result familiar. 2

83

x

x

A shortcut

When the divisor is linear, we can use something called synthetic division to find a quick answer.

What was our answer before? How does it compare to these numbers?

2

1232 23

x

xxx

Shortcut again

Try it with this one. Watch for spacers!

2

83

x

x