ZEROS AND

END BEHAVIOR

The “zero” of a function is just the value at which a function touches the x-axis.

It is easy to find the roots of a polynomial when it is in factored form!

2 2 15 ( 3)( 5)x x x x

(x - 3) and (x + 5) are factors of the polynomial.

Factored Polynomial

(x - 3) and (x + 5) are factors of the polynomial.

(x - 3)(x + 5) = 0 (we want to know where the polynomial crosses the x-axis)

So (x – 3) = 0 and (x + 5) = 0

The zeros are x = 3, x = -5

Practice: Find the roots of the following factored polynomials.

1. y = (x-2)3(x+3)(x-4)

2. y = (x-5)(x+2)3(x-14)2

3. y = (x+3)(x-15)4

4. y = x2(x+6)(x-6)

Sometimes the polynomial won’t be factored!

Ex. xxxy 623

2nd → TRACE (CALC) → 2: zero

Choose a point to the left of the zero.Then press ENTER.

This arrow indicates that you’ve chosen a

point to the left of the zero.

Choose a point to the rightof the zero.Then press ENTER.

This arrow indicates that you’ve chosen a point to the right of

the zero.

Press ENTER one more time!

Find the zeros of the following polynomials:

13722456317

4562810

234

234

xxxxy

xxxxy

Solutions

4,713722456317

5,1,34562810

234

234

yxxxxy

yxxxxy

End Behavior

The end behavior of a graph describes the far left and the far right portions of the graph.

We can determine the end behaviors of a polynomial using the leading coefficient and the degree of a polynomial.

First determine whether the degree of the polynomial is even or odd.

Next determine whether the leading coefficient is positive or negative.

532)( 2 xxxf degree = 2 so it is even

Leading coefficient = 2 so it is positive

DegreeEven Odd

Lead

ing

Coeffi

cien

t

+

−

High→High Low→High

Low→Low High→Low

Find the end behavior of the following polynomials.

952)( a. 3 xxxf

3624)( b. 24 xxxxf

xxxxf 234)( c. 25

4323)( d. 234 xxxxxf