1. Real Zeros of Polynomial Functions Long Division and Synthetic Division

2. Warm-up 1-2 Factor and Simplify each expression.

3. Warm-up 3-4 Factor each expression completely.

4. Warm-up 5-6 Multiply (FOIL) each expression.

5. Warm-up 7. Evaluate

6. Warm-up – Answer #7

7. Multiplication Equations Can be written as two or more division equations.

8. Think about this in terms of Area If I have a box lid with a length of x 2 +5x+6 and a width of x-2, what is the area of the box?

9. Dividing by a Monomial Properties of exponents are used to divide a monomial by a monomial and a polynomial by a monomial.

10. Division by a Polynomial A polynomial can be divided by a divisor of the form x-r by using long division or a shortened form of long division called synthetic division. Long division is similar to long division of real numbers. This method works for dividing by any type of term. Synthetic division uses only the coefficients of the terms in the process. To use synthetic division your divisor must be linear!

11. Polynomials Consider the graph of f(x) = 6x 3 - 19x 2 + 16x - 4 Notice that the graph appears to cross the x-axis at 2. From the graph we know that x = 2 appears to be a zero meaning f(2) = 0. If x = 2 is a zero then we also know f(x) has a factor of (x-2). How many additional zeros does the graph have? This means that there exists a 2nd degree polynomial q(x) such that To find q(x) without a calculator we will use long or synthetic division. What is the degree of the polynomial? How many real zeros does the graph have?

12. Division Rules Dividend equals divisor times quotient plus remainder.

13. Long Division Use a process similar to long division of whole numbers to divide a polynomial by a polynomial. Leave space for any missing powers of x in the dividend. Write the remainder as you would with whole numbers. (Remainder over the divisor)

14. Example 1 – Long Division 1.Divide the “x” into 6x 3 2.Multiply the 6x 2 by the divisor (x-2) 3.Subtract 4.Bring down the “16x” Repeat: 1.Divide “x” into 7x 2 2.Multiply (-7x) times (x-2) 3.Subtract 4.Bring Down the 4 Repeat… 6x 2 -7x +2 Remainder is Zero

15. Example 2 – Long Division Subtract!! Subtract!! Subtract!!

16. The Remainder Theorem If a polynomial f(x) is divided by x – k, the remainder is r = f(k).

17. Example 2 Concluded Using the remainder theorem we know that f(-1/3) = - 7 and since the remainder is not 0, x = -1/3 is not a root of the function.

18. Long Division Practice 1-3

19. Answers Practice 1-3 P1 P2 P3

20. Synthetic Division Shortened form or “short cut” of Long division Must have a linear divisor Only uses coefficients not variables Zeros must be included to hold the place of any power of x that is missing.

21. Synthetic Division The pattern for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.) 1.Write the coefficients of the dividend in a upside-down division symbol. 2.Take the zero of the divisor, and write it on the left. 3.Carry down the first coefficient. 4.Multiply the zero by this number. Write the product under the next coefficient. 5.Add. 6.Repeat as necessary

22. Example 3 - Synthetic Division Divide x 4 – 10x 2 – 2x + 4 by x + 3 10-10-24 -3 1 +9 3 1 -3 1 R Remainder C Constant x x2x2 x3x3 Multiply Add Coefficients →

23. Example 4 – Synthetic Division 27-4-27-18 +2 2 4 11 22 18 36 9 18 0 R C X X2X2 X3X3

24. Example 4 - Synthetic Division Is (x-2) a factor of the function? Why or Why not? What is the value of f(2)? Where would you find this point on a graph? Yes Definition of Division f(2) = 0 On the x-axis at x = 2 It is an x-intercept or root of the function.

25. Synthetic Division Practice 4-6

26. Answers Practice 4-6 P4 P5 P6 x 2 x C R x 3 x 2 x C R x 2 x C R