1. STD 3: To divide polynomials using long division and synthetic division and the rational root theorem Warm Up 1.432 /15

2. Std Division of Polynomials Use long division to divide x 2 + 4x + 3 by x + 1

3. Std Division of Polynomials Use long division to divide 3x 2 - 5x + 6 by x + 2

4. Std Division of Polynomials Use long division to divide x 3 - 1 by x + 1

5. Std Division of Polynomials You try: 1.x 2 – 5x + 2 by x – 1 2. 2x² – 3 by 2x + 3

6. Obj: To divide polynomials using synthetic division and the rational root theorem Synthetic division is an easier way to divide polynomials. (kind of like short division) Still follow the rules of putting polynomials in descending order and make sure you have a placeholder for each degree. However, you only need to write the coefficient for each term. Ex: 3x⁴ + 2x³ - 6x² + x - 4 You write 3 2 -6 1 -4

7. Obj: To divide polynomials using synthetic division and the rational root theorem Also the divisor, you write using the zero. Example (x + 3) you write -3. Example: (3x⁴ + 2x³ - 6x² + x – 4) ÷ (x + 3)

8. Obj: To divide polynomials using synthetic division and the rational root theorem Example: (x³ - 3x² - 5x – 25) ÷ (x - 5)

9. Obj: To divide polynomials using synthetic division and the rational root theorem Determine if the binomial is a factor of x³ + 4x² + x – 6 a) x + 1b) x + 2

10. Obj: To divide polynomials using synthetic division and the rational root theorem Use synthetic division to completely factor each polynomial function y= x³ + 2x² - 5x – 6 (x + 1)

11. Obj: To divide polynomials using synthetic division and the rational root theorem Remainder Theorem If a polynomials P(X) of degree n ≥ 1 is divided by (x – a), where a is a constant, then the remainder is P(a). Example: P(x) = x³ + 4x² - 8x – 6; a = -2

12. Obj: To divide polynomials using synthetic division and the rational root theorem Rational Root Theorem- If p/q is in simplest form and is a rational root of the polynomial equation a n xⁿ + a n-1 xⁿ⁻ 1 + … + a 1 x + a 0 = 0 with integer coefficients, then p must be a factor of a 0 and q must be a factor of a n.

13. Obj: To divide polynomials using synthetic division and the rational root theorem Example: x³ + x² - 3x – 3 = 0

14. Obj: To divide polynomials using synthetic division and the rational root theorem Find the roots of 2x³ - x² + 2x – 1 = 0

15. Obj: To divide polynomials using synthetic division and the rational root theorem Irrational Roots and complex number roots always occur in conjugate pairs!!! Find a third degree polynomial equation with rational coefficients that has the given numbers as roots. Ex: 1 and 3iEx: 3 + i and -3