1. ALGEBRA 3
Polynomial Division

2. How? (x3 + 5x2 + 7x + 2) ÷ (x + 2) x2 x + 2 ) x3 + 5x2 + 7x + 2

3. How? (x3 + 5x2 + 7x + 2) ÷ (x + 2) x2 + 3x x + 2 ) x3 + 5x2 + 7x + 2

4. Nearly there! (x3 + 5x2 + 7x + 2) ÷ (x + 2) x2 + 3x + 1
0 + 0

5. What does this mean?
Because we have ended up with zero at the bottom it means that (x + 2) divided exactly into our polynomial, with no remainder.
Anything else left at the bottom would be the remainder.

6. Why would you bother?
This method can help to factorise polynomials that have a higher order than we usually deal with.
For example, once you have divided a cubic (order 3) polynomial by a linear (order 1) factor, you will be left with a quadratic and you should know how to factorise those!

7. Does this work with our example?
(x3 + 5x2 + 7x + 2) ÷ (x + 2)
Because (x + 2) left no remainder we can write:
(x3 + 5x2 + 7x + 2) = (x + 2)(x2 + 3x + 1)
How can we check this is true?

8. Does this work with our example?
(x3 + 5x2 + 7x + 2) = (x + 2)(x2 + 3x + 1)
Can you factorise (x2 + 3x + 1)?
Try this example instead:
(x3 + 2x2 – x – 2) ÷ (x + 2)