1. 3.6 Polynomial Functions Part 2
Chapter 3
3.6 Polynomial Functions
Part 2

2. Objectives: SWBAT:
Find the roots of a polynomial function analytically given at least one of the roots.
HW: p. 223: (even); (even)

3. Remainders and Factors:
If the remainder is 0 when one polynomial is divided by another polynomial, the divisor and the quotient are factors of the polynomial.

4. Remainder Theorem:
The remainder when f(x) is divided by (x – c) is f(c).

5. Students Try!!!

6. Factor Theorem:

7. Fundamental Polynomial Connections:
Let f(x) be a polynomial. If r is a real number that satisfies any of the following conditions, then r satisfies all the statements:
r is a zero of the function f.
r is an x-intercept of the graph of the function f.
x = r is a solution, or root, of the equation f(x) = 0.
x – r is a factor of the polynomial f(x).

8. Example: One of the zeros of f(x) = 3x3 – 2x2 – 7x – 2, is 2. Find:
all the zeros of f
the x-intercepts of the graph of f
the solutions to 3x3 – 2x2 – 7x – 2 = 0
the linear factors with real coefficients of
3x3 – 2x2 – 7x – 2

9. Students Try!!! Answers: the zeros of f 2, –⅓, –1
the x-intercepts of the graph of f
(2, 0), (–⅓, 0), (–1, 0)
the solutions to 3x3 – 2x2 – 7x – 2 = 0
x = 2, –⅓, –1
the linear factors with real coefficients of
3x3 – 2x2 – 7x – 2
(x – 2)(x + ⅓)(x +1)

10. The factored form of the original polynomial is:

11. Number of Zeros A polynomial of degree n has at most n distinct zeros.
Ex: State the maximum number of distinct real zeros of f:
f(x) = 4x7 – 2x6 – 7x3 – 2x + 4
Answer: 7

12. Students Try!!!
Find a polynomial function g of degree 4 such that the zeros of g are 0, -1, 2, -3, and g(3) = Leave the polynomial in factored form.
g(x) = a(x – 0)(x + 1)(x – 2)(x + 3)
g(x) = a(x)(x + 1)(x – 2)(x + 3)
288 = a(3)(3 + 1)(3 – 2)(3 + 3)
288 = a(3)(4)(1)(6)
288 = 72a
a = 4
g(x) = 4(x)(x + 1)(x – 2)(x + 3)