Aim: How do we find the zeros of polynomial functions? Do Now: A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?

General Features of a Polynomial Function DEFINITION: Polynomial Function Let n be a nonnegative integer and let a0, a1, a2, . . . an-1, an be real numbers with an ≠ 0. The function given by f(x) = anxn + an - 1xn - 1 + . . . . + a2x2 + a1x + a0 is a polynomial function of degree n. The leading coefficient is an. The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Describe some basic characteristics of this polynomial function: Continuous no breaks in curve Smooth no sharp turns discontinuous sharp turn NOT POLYNOMIAL FUNCTIONS

General Features of a Polynomial Function Standard Form Degree Leading Coefficient Cubic term Quadratic term Constant term Linear term Polynomial of 4 terms

General Features of a Polynomial Function Simplest form of any polynomial: y = xn n > 0 When n is even looks similar to x2 When n is odd looks similar to x3 The greater the value of n, the flatter the graph is on the interval [ -1, 1].

Transformations of Higher Degree Polys If k and h are positive numbers and f(x) is a function, then f(x ± h) ± k shifts f(x) right or left h units shifts f(x) up or down k units f(x) = (x – h)3 + k - cubic f(x) = (x – h)4 + k - quartic ex. f(x) = (x – 4)4 – 2

Zeros of Polynomial Functions The zero of a function is a number x for which f(x) = 0. Graphically it’s the point where the graph crosses the x-axis. For polynomial function f of degree n, the function f has at most n real zeros the graph of f has at most n – 1 relative extrema (relative max. or min.). Ex. Find the zeros of f(x) = x2 + 3x f(x) = 0 = x2 + 3x = x(x + 3) x = 0 and x = -3 How many roots does f(x) = x2 + 1 have?

Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number plane. Degree of polynomial Function Zeros 1st n = 1 f(x) = x – 3 x = 3 2nd n = 2 f(x) = x2 – 6x + 9 = (x – 3)(x – 3) x = 3 and 3rd n = 3 f(x) = x3 + 4x = x(x – 2i)(x + 2i) x = 0, x = 2i, x = -2i 4th n = 4 f(x) = x4 – 1 = (x – 1)(x + 1)(x – i)(x + i) x = 1, x = -1, x = i, x = -i repeated zero

Finding Zeros Find the zeros of f(x) = x3 – x2 – 2x Has at most 3 real roots Has 2 relative extrema f(x) = 0 = x3 – x2 – 2x = x(x2 – x – 2) = x(x – 2)(x + 1) x = 0, x = 2 and x = -1

Finding a Function Given the Zeros Write a quadratic function whose zeros (roots) are -2 and 4. x = -2 x = 4 reverse the process used to solve the quadratic equation. x + 2 = 0 x – 4 = 0 (x + 2)(x – 4) = 0 x2 – 2x – 8 = 0 x2 – 2x – 8 = f(x) Find a polynomial function with the following zeros: -2, -1, 1, 2 f(x) = (x + 2)(x + 1)(x – 1)(x – 2) f(x) = (x2 – 4)(x2 – 1) = x4 – 5x2 + 4

Multiplicity Find the zeros of f(x) = x4 + 6x3 + 8x2. f(x) = x4 + 6x3 + 8x2 A multiple zero has a multiplicity equal to the numbers of times the zero occurs.

Regents Prep The graph of y = f(x) is shown at right. Which set lists all the real solutions of f(x) = 0? {-3, 2} {-2, 3} {-3, 0, 2} {-2, 0, 3}

Find the zeros of f(x) = 27x3 + 1. Model Problem Find the zeros of f(x) = 27x3 + 1. Factoring Difference/Sum of Perfect Cubes u3 – v3 = (u – v)(u2 + uv + v2) u3 + v3 = (u + v)(u2 – uv + v2)

Model Problem Find the zeros of f(x) = 27x3 + 1.

Polynomial in Quadratic Form Find the zeros = 0 2 u’s – 4 zeros

Finding zeros by Factoring by Groups Find the roots of the following polynomial function. f(x) = x3 – 2x2 – 3x + 6 x3 – 2x2 – 3x + 6 = 0 Group terms (x3 – 2x2) – (3x – 6) = 0 Factor Groups x2(x – 2) – 3(x – 2) = 0 Distributive Property (x2 – 3)(x – 2) = 0 x2 – 3 = 0; Solve for x x – 2 = 0; x = 2

Regents Prep Factored completely, the expression 12x4 + 10x3 – 12x2 is equivalent to