1. 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?

2. 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 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

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

4. 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].

5. 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

6. 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?

7. 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

8. 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

9. 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

10. 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.

11. 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}

12. 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)

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

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

15. 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

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