Roots & Zeros of Polynomials I

How the roots, solutions, zeros, x-

intercepts and factors of a polynomial

function are related.

I. Review of PolynomialsA. Expressions vs. Equations

A Polynomial Expression can be a monomial or a sum of monomials. The Polynomial Expressions that we are discussing today are in terms of one variable.

In a Polynomial Equation, two polynomials are set equal to each other.

B. Zero Product Property

For all numbers a and b, if ab = 0, then

a = 0, b = 0, or both a and b equal 0

C. Solving a Polynomial Equation

The only way that x2 +2x - 15 can = 0 is if x = -5 or x = 3

Rearrange the terms to have zero on one side:

2 22 15 2 15 0x x x x Factor:

( 5)( 3) 0x x

Set each factor equal to zero and solve: ( 5) 0 and ( 3) 0

5 3

x x

x x

D. Factors, Roots, Zeros

y x2 2x 15For our Polynomial Function:

The Factors are: (x + 5) & (x - 3)

The Roots/Solutions are: x = -5 and 3

The Zeros are at: (-5, 0) and (3, 0)

E. Graph of a Polynomial Function

Here is the graph of our polynomial function:

The Zeros of the Polynomial are the values of x when the polynomial equals zero. In other words, the Zeros are the x-values where y equals zero.

These are also the roots and the x-intercepts.

y x2 2x 15

II. Finding RootsA. Fundamental Theorem of Algebra

Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers.

A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers.

COROLLARY:

B. Real/Imaginary RootsIf a polynomial has ‘n’ complex roots will its graph have ‘n’ x-intercepts?

In this example, the degree n = 3, and if we factor the polynomial, the roots are x = -2, 0, 2. We can also see from the graph that there are 3 x-intercepts.

y x3 4x

Real/Imaginary RootsJust because a polynomial has ‘n’ complex roots doesn’t mean that they are all Real!

y x3 2x2 x 4In this example, however, the degree is still n = 3, but there is only one Real x-intercept or root at x = -1, the other 2 roots must have imaginary components.

C. Finding Roots of a Polynomial using ZPP and factoring.

We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring.

Ex 1: Some are easier to factor than others!

f (x) x3 4x

x(x2 4)

x(x 2)(x 2)

The roots are: 0, -2, 2

Ex 2: Some are harder to factor. Remember Grouping?

2, 2x x = 5

Don’t forget your other factoring and solving methods, including GCF, memorized patterns and the quadratic formula.

D. Complex Roots

For now, you will only have to find the real roots. We’ll worry about complex roots later this semester. The directions in your homework might ask you to find all roots, but you only need to find the real roots.

HW #13: Page 210: 31, 39-47, 49-50 # and location of real roots only

Finding Roots/Zeros of Polynomials

We use the Fundamental Thm. Of Algebra to predict the nature of the roots of a polynomial.

We use skills such as factoring, polynomial division and the quadratic formula to find the zeros/roots of polynomials.