1. Division of Polynomials The Long and Short of it.

2. Review of Vocabulary Factor root, zero, solution, x-intercept dividend ÷ divisor = quotient Example: in 12 ÷ 3 = 4: 12 is the dividend 3 is the divisor 4 is the quotient remainder

3. 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.

4. 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 : Purplemath I will demonstrate a quicker way also.

5. Example 1 Divide x 2 – 9 x – 10 by x + 1 The process is similar to long division of numbers as you learned in elementary school.

6. 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 x 2 of the dividend. If I divide the leading x 2 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 x 2 underneath: Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath:

7. 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...subtract...and then I add down. The first term (the x 2 ) 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:

8. 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:

9. finally The answer is x-10. That means x 2 – 9 x – 10 divide by x + 1 = x-10. In another words: x 2 – 9 x – 10 = ( x + 1)(x-10) One more way to say: x-10 is a factor of x 2 – 9 x – 10

10. Why is division useful?

11. 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.

12. 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.

13. 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.

14. 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.

15. Example 1

16. -6 1 16 59 -6 1 -6 10 -60 -1 6 0 x 2 + 10x - 1 R=0 Opposite of number in divisor

17. Therefore

18. Example 2

19. -3 4 18 12 -18 4 -12 6 -18 -6 18 0 4x 2 + 6x - 6 R=0 Opposite of number in divisor

20. 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.

21. -7 1 6 -15 -52 1 -7 -1 7 -8 56 4 Opposite of number in divisor

22. Ex 3 Continued: