1. Synthetic Division Objective: To use synthetic division to determine the zeros of a polynomial function.

2. Review Determine the zeros of f(x) = x 2 – 3x – 10 by factoring. –f(x) = (x – 5)(x + 2) –(x – 5)(x + 2) = 0 –x – 5 = 0 or x + 2 = 0 –x = 5 or x = -2 –the zeros are {-2, 5}  Use the zero product property

3. Review Solve by factoring 27x 3 – 8 = 0 –(3x – 2)(9x 2 + 6x + 4) = 0 –3x – 2 = 0 or –3x = 2 or – or –The solutions are  Use the zero product property and the quadratic formula

4. Review Solve by factoring x 3 – 3x 2 – 6x + 18 = 0 –(x 3 – 3x 2 ) + (-6x + 18) = 0 –x 2 (x – 3) – 6(x – 3) = 0 –(x – 3)(x 2 – 6) = 0 –x – 3 = 0 or x 2 – 6 = 0 –x = 3 or x 2 = 6 –The solutions are  Use the zero product property and square roots

5. Synthetic Division Find the zeros of f(x) = x 3 + x 2 – 10x – 6 given that one of the zeros is 3. This cubic polynomial must have 3 zeros. It could have 3 real zeros or 1 real and 2 imaginary zeros. As this polynomial cannot be factored by grouping, we need to have a different method. That method is called synthetic division.

6. Synthetic Division Synthetic division is a method of dividing a polynomial by a binomial. It is used in place of long division. When doing synthetic division, you only use the coefficients of the variables. Because of this, you must insure that the polynomial is in standard form and there are no missing terms.

7. Synthetic Division The process for synthetic division is as follows: –Set up the division problem using the coefficients of the terms as the dividend and the given real zero as the divisor. –Bring down the first coefficient. Multiply this number by the zero, place it under the second coefficient and ADD (this is different than regular long division). Continue. –When you find the quotient, put the variables back in. –If necessary, use synthetic division again to further reduce (depress) the polynomial.

8. Example: Find the zeros of f(x) = x 3 + x 2 – 10x – 6 given that one of the zeros is 3. –Step 1: Set up the problem The division problem is set up upside-down in comparison to a regular long division problem. The quotient will be BELOW the problem.

9. Example: Step 2: Drop down the first coefficient. Then multiply it by the zero and place the answer under the second coefficient. ADD the two numbers and place the answer in the quotient row.

10. Example: Step 3: Continue for the remaining coefficients. IMPORTANT!! The last number in the quotient row is the remainder. If the remainder is NOT zero, then the given number is not a zero of the function.

11. Example: Step 4: Place the variables into the quotient. It is easiest to work from RIGHT to LEFT remembering that the last number is the remainder and the on to its left is the constant.

12. Example The original polynomial f(x) = x 3 + x 2 – 10x – 6 can now be written in factored form as: –f(x) = (x – 3)(x 2 + 4x + 2) Once you have depressed the original polynomial to a quadratic, you can solve for the remaining zeros using factoring, square roots or the quadratic formula.

13. Example We were given one zero of the original polynomial (3), we can now find the other two using the quadratic formula. –f(x) = (x – 3)(x 2 + 4x + 2) The zeros of this function are

14. Example Find the zeros of the function f(x) = 2x 3 + 7x 2 + 11x + 10 given that (x + 2) is one factor. –Step 1: Set up the problem Remember that the divisor is the ZERO of the function

15. Example –Step 2: Drop the first coefficient, then multiply and add. –Step 3: Replace the variables

16. Example –Step 4: Find the remaining zeros –f(x) = (x + 2)(2x 2 + 3x + 5) The zeros of this function are

17. Example Find the zeros of the function f(x) = 3x 3 – 19x – 24. Use your calculator to find one real zero. –Step 1: Graph the function on your calculator and find one real zero. There is a zero at x = 3. –Step 2: Set up the problem Because the quadratic (x 2 ) term is missing, use a 0 as a placeholder for its coefficient.

18. Example –Step 3: Drop the first coefficient, then multiply and add. –Step 4: Replace the variables

19. Example –Step 5: Find the remaining zeros –f(x) = (x – 3)(3x 2 + 9x + 8) The zeros of this function are

20. Example Determine if (x – 5) is a factor of f(x) = 3x 4 – 9x 3 – 34x 2 + 16 using synthetic division. –To determine this, use synthetic division and see if the remainder is 0.

21. Example Use the potential zero (-5) as the divisor and put a placeholder of 0 in for the missing linear term. Because the remainder is NOT 0, 5 is not a zero of the function and (x – 5) is not a factor.

22. Example: Even though (x – 5) is not a factor of this function, we can still write the quotient of this division problem. –Put the variables back in.

23. Example (3x 4 – 9x 3 – 34x + 16) ÷ (x – 5) = The remainder is written as a fraction, just like in traditional long division.