1. Pre-Calculus Section 2.3 Synthetic Division
Pre Calc Sec 2-3
Pre-Calculus Section 2.3 Synthetic Division

2. Synthetic Division
Use when dividing by linear binomial (x - k) polynomial must be in order put in zeros for missing terms multiply then add write answer with variables in it

3. Use synthetic division to divide. 1. - 10x2 - 2x + x4 + 4 by x + 3

4. Remainder Theorem
If a polynomial f(x) is divided by x - k, the remainder is r = f(k) This means that when we use synthetic division the remainder is equal to the value of the divisor put into the function.

5. Use the Remainder Theorem to evaluate the following function at x = - 2 2. f(x) = 3x3 + 8x2 + 5x - 7

6. Factor Theorem
A polynomial f(x) has a factor (x - k) if and only if f(k) = 0, which means that the remainder = 0 Means that when we use synthetic division if the remainder is equal to 0 we have found a factor / zero / solution of the function.

7. Show that (x - 2) and (x + 3) are factors of 3
Show that (x - 2) and (x + 3) are factors of 3. f(x) = 2x4 + 7x3 - 4x2 - 27x - 18

8. Use synthetic division to show that x is a solution, factor completely, and list all the real zeros of the function. 4. f(x) = x3 - 7x + 6 = 0 x = 2

9. 5. f(x) = 2x3 - 15x2 + 27x - 10 = x = ½

10. Verify the given factors of the function f
Verify the given factors of the function f. Find the remaining factors of f, write the complete factorization of f, and list all real zeros of f. 6. f(x) = x4 - 4x3 - 15x2 + 58x - 40 factors: (x - 5) (x + 4)

12. Steps to Find Rational Zeros
Rational Zero Test
Used to find the zeros of polynomials that are of a degree of 3 or higher.
Steps to Find Rational Zeros
1st - Use the graphing calculator – list the zeros
2nd - Use synthetic division to prove all of the zeros
3rd – Write as factors if directions state

13. 7. Find the rational zeros of f(x) = 2x3 + 3x2 - 8x + 3

15. 8. Find all the real zeros of f(x) = 10x3 - 15x2 - 16x + 12

17. 9. Find all the real zeros of f(x) = x4 - x3 - 2x - 4

19. 10. Find all the real zeros of f(x) = 4x5 + 12x4 - 11x3 - 42x2 + 7x + 30

21. Verify the given factors of the function f and find the remaining factors of f. Then write the complete factorization of f. 11. 𝑓 𝑥 =2𝑥 3 + 𝑥 2 −5𝑥 + 2 𝑓𝑎𝑐𝑡𝑜𝑟 (𝑥+2)

23. Verify the given factors of the function f and find the remaining factors of f. Then write the complete factorization of f. 12. 𝑓 𝑥 = 𝑥 4 − 4𝑥 3 − 15 𝑥 2 +58𝑥 − 40 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑥+4 , (𝑥 − 5)