The “zero” of a function is just the
value at which a function touches the
x-axis.
Factored Polynomial
It is easy to find the roots of a polynomial when it is in
factored form!
x  2 x  15  ( x  3)( x  5)
2
(x - 3) and (x + 5) are factors of the 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. y  x 3  x 2  6 x
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:
y  x  10 x  28 x  6 x  45
4
3
2
y  x  17 x  63 x  245 x  1372
4
3
2
Solutions
y  x  10 x  28 x  6 x  45
4
3
2
y   3 , 1,  5
y  x  17 x  63 x  245 x  1372
4
3
2
y  7, 4
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.
2
f ( x)  2x  3x  5
degree = 2 so it is even
Next determine whether the leading coefficient
is positive or negative.
Leading coefficient = 2 so it is positive
Leading Coefficient
Degree
Even
Odd
High→High
Low→High
Low→Low
High→Low
+
−
Find the end behavior of the
following polynomials.
a. f ( x )   2 x  5 x  9
3
b. f ( x )  4 x  2 x  6 x  3
4
2
c. f ( x )  4 x  3 x  2 x
5
2
d. f ( x )   3 x  2 x  x  3 x  4
4
3
2