2.2 Polynomial Function of Higher Degrees Zero of the function Intermediate Value Theorem

Polynomial functions “Continuous” meaning no holes or breaks Polynomial graphs have no sharp turns

Would a absolute value function be a polynomial?

Would a absolute value function be a polynomial? Not a polynomial function

f(x) = xn is a Power function If n is even If n is odd The larger n, the flatter it is by the origin.

Ending behavior as x → ∞ If n is even and an > 0, as x → - ∞

Ending behavior as x → ∞ If n is odd and an > 0, as x → - ∞

Zeros of a Polynomial function Numbers in the domain that make f(x) = 0. We can the zeros with the factoring, the Quadratic formula or Synthesis Division. Find the zeros of g(x) = x 4 – 2x3 – 63x2 What is another name for a zero of a function?

Zeros of a Polynomial function Numbers in the domain that make f(x) = 0. We can the zeros with the factoring, the Quadratic formula or Synthesis Division. Find the zeros of g(x) = x 4 – 2x3 – 63x2 What is another name for a zero of a function? Roots (lets call r) : so (x – r)(x – r)(x – r)… depends on degrees

g(x) = x 4 – 2x3 – 63x2

g(x) = x 4 – 2x3 – 63x2 It factors into x2(x + 7)(x – 9) Set each to zero x2 = 0 x = 0 (d.r.) x + 7 = 0 x = - 7 x – 9 = 0 x = 9 How would this help you graph?

What is the ending behavior? - 7 0 9

What is the ending behavior? - 7 0 9

What is the ending behavior? What is happening between the zeros? put numbers in g(x) between the zeros to find out it the function is positive or negative - 7 0 9

Testing for positive or negative Between – 7 and 0 lets use – 1 Between 0 and 9 lets use 1 g( -1) = (-1)2(- 1 + 7)(- 1 – 9) = (1)(6)(-10) negative g(1) = (1)2(1 + 7)(1 – 9) = (1)(8)(-8) In both cases the graph is below the x axis.

What is the ending behavior? - 7 0 9 It gives the idea of what the graph should look like.

The Intermediate Value Theorem Let a and b be real numbers such that a < b. If “f” is a polynomial function such that f(a)≠ f(b), then in the interval [a,b] ,“f” takes on every value between f(a) and f(b). What does this do?

The Intermediate Value Theorem Let a and b be real numbers such that a < b. If “f” is a polynomial function such that f(a)≠ f(b), then in the interval [a,b] ,“f” takes on every value between f(a) and f(b). What does this do? When f(x) goes from positive to negative or negative to positive, there exist one zero.

Find the intervals ( of 1 unit) where there is a zero of the function f(x) = x4 + x3 – 7 f(- 3) = (- 3)4 + (-3)3 – 7 = 81 – 27 – 7 f(-3) = 47 f(-2) = 1 f(- 1) = -7 zero between – 2 and - 1 f(0) = -7 f(1) = - 5 f(2) = 17 zero between 1 and 2

Proof of the existence of √2 f(x) = x2 – 2 f(1) = (1)2 – 2 = - 1 f(2) = (2)2 – 2 = 2 There is a zero between [ 1, 2] Thus 0 = x2 – 2; 2 = x2 x = √2

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