1. Divide Polynomials Objectives: 1.To divide polynomials using long division and synthetic division

2. Warm-Up Use long division to divide 5 into 3462. - - -

3. Warm-Up - - - Divisor Dividend Quotient Remainder

4. Warm-Up Use long division to divide 5 into 3462. Dividend Divisor Quotient Remainder Divisor

5. Remainders divides evenly If you are lucky enough to get a remainder of zero when dividing, then the divisor divides evenly into the dividend factor This means that the divisor is a factor of the dividend For example, when dividing 3 into 192, the remainder is 0. Therefore, 3 is a factor of 192.

6. Vocabulary QuotientRemainder DividendDivisor Divides EvenlyFactor

7. Objective 1a You will be able to divide polynomials using long division

8. Dividing Polynomials long division Dividing polynomials works just like long division. In fact, it is called long division! Before you start dividing: Make sure the divisor and dividend are in standard form If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder

9. Dividing Polynomials long division Dividing polynomials works just like long division. In fact, it is called long division! Before you start dividing: If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder

10. Exercise 1 Divide x + 1 into x 2 + 3 x + 5 Line up the first term of the quotient with the term of the dividend with the same degree. How many times does x go into x 2 ? Multiply x by x + 1 -- Multiply 2 by x + 1 --

11. Exercise 1 Divide x + 1 into x 2 + 3 x + 5 -- -- Divisor Dividend Quotient Remainder

12. Exercise 1 Divide x + 1 into x 2 + 3 x + 5 Divisor Dividend Quotient Remainder Divisor

13. Exercise 2 Divide 3 x 4 – 5 x 3 + 4 x – 6 by x 2 – 3 x + 5

14. Exercise 3 In a polynomial division problem, if the degree of the dividend is m and the degree of the divisor is n, what is the degree of the quotient?

15. Exercise 4 Divide using long division.

16. Exercise 5 Use long division to divide x 4 – 10 x 2 + 2 x + 3 by x – 3

17. You will be able to divide polynomials using synthetic division

18. Synthetic Division When you divisor is of the form x  k, where k is a constant, then you can perform the division quicker and easier using just the coefficients of the dividend. synthetic division This is called fake division. I mean, synthetic division.

19. Synthetic Division Synthetic Division Synthetic Division (of a Cubic Polynomial) To divide ax 3 + bx 2 + cx + d by x – k, use the following pattern. kabcd a ka = Add terms = Multiply by k Coefficients of Quotient (in decreasing order) Remainder

20. Synthetic Division Synthetic Division Synthetic Division (of a Cubic Polynomial) To divide ax 3 + bx 2 + cx + d by x – k, use the following pattern. kabcd a ka = Add terms = Multiply by k You are always adding columns using synthetic division, whereas you subtracted columns in long division.

21. Synthetic Division Synthetic Division (of a Cubic Polynomial) To divide ax 3 + bx 2 + cx + d by x – k, use the following pattern. Synthetic Division You are always adding columns using synthetic division, whereas you subtracted columns in long division. k can be positive or negative. If you divide by x + 2, then k = -2 because x + 2 = x – (-2). Add a coefficient of zero for any missing terms!

22. Exercise 6 Use synthetic division to divide x 4 – 10 x 2 + 2 x + 3 by x – 3

23. Exercise 7 Divide 2 x 3 + 9 x 2 + 4 x + 5 by x + 3 using synthetic division

24. Exercise 8 Divide using long division.

25. Exercise 9 Given that x – 4 is a factor of x 3 – 6 x 2 + 5 x + 12, rewrite x 3 – 6 x 2 + 5 x + 12 as a product of two polynomials.

26. Exercise 10 The volume of the solid is 3 x 3 + 8 x 2 – 45 x – 50. Find an expression for the missing dimension. x + 5 ? x + 1

27. Exercise 11 Use long division to divide 6 x 4 – 11 x 3 + 14 x 2 – 3 x – 1 by 2 x – 1. Then figure out a way to perform the division synthetically.

28. Divide Polynomials Objectives: 1.To divide polynomials using long division and synthetic division