1. 4.3 – The Remainder and Factor Theorems

2. “I can use the Remainder and Factor Theorems to find factors of polynomials”
REMAINDER THEOREM: If a polynomial P(x) is divided by x – r, the remainder is a constant P(r), and P(x) = (x – r) ∙ Q(x) + P(r), where Q(x) is a polynomial with degree one less than the degree of P(x).

3. “I can use the Remainder and Factor Theorems to find factors of polynomials”
REMAINDER THEOREM: Example: Let P(x) = 2x2 + 3x – 8, first divide P(x) by x – 2. Next, find P(2)
First Answer: 2a + 7 with a remainder of 6
Second Answer: P(2) = 6

4. “I can use the Remainder and Factor Theorems to find factors of polynomials”
FACTOR THEOREM: The binomial x – r is a factor of the polynomial P(x) if and only if P(r) = 0. Example: Determine if x – 1 is a factor of P(x) = 2x3 – 3x2 + x.
Answer: yes, because P(1) = 0

5. “I can divide polynomials using synthetic division.”
A short-cut to long division.
Only works when dividing by a binomial of the form x – r.
Example: Divide –y6 + 4y4 + 3y2 + 2y by y + 2.
Answer: -y5 + 2y4 + 3y – 4 remainder 8 *Note: substitute zero place holders since all exponents are not accounted for in the problem!

6. “I can factor polynomials using the Factor Theorem and synthetic division.”
RATIONAL ROOTS THEOREM:
Let a0xn + a1xn-1 + … + an-1x + an = 0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0.
Example:
All possible rational roots of 6x3 + 11x2 – 3x – 2 = 0 are
p = +1, q = +1, , + 6 
p/q = +1, +1/2, +1/3, +1/6, +2, and +2/3

7. Determine the binomial factors of x3 – 2x2 – 13x – 10.
First, find a rational root by using the Rational Root and Factor Theorems.
Second, use synthetic division with the rational root you found in the first part.
Third, repeat steps 1 and 2 until you have a quadratic equation left to factor.

8. Determine the binomial factors of x3 – 2x2 – 13x – 10.
First, find a rational root by using the Rational Root and Factor Theorems.
All possible roots: (all pos and neg) 1, 2, 5, , -2, and 5 work

9. Determine the binomial factors of x3 – 2x2 – 13x – 10
Second, use synthetic division with the rational root you found in the first part.
Just divide 1 time with the first root you found from -1, -2, and 5.

10. Determine the binomial factors of x3 – 2x2 – 13x – 10
Third, repeat steps 1 and 2 until you have a quadratic equation left to factor.
Final answer (x + 1)(x + 2)(x – 5)

11. Homework
4.3 p226 #29 – 34, 37, 38, 49