Finding zeros of a function

An SAT Math micro-topic under Polynomial and other nonlinear graphs (Advanced Math). Free to read — no account needed.

A zero of a function

f

is any input

x

for which

f(x) = 0

. Because the output is zero there, the graph of the function crosses or touches the

x

-axis at each zero.

The figure shows both kinds. At one zero the curve passes straight through the axis (a simple zero); at another it just touches and turns back — a double zero, which happens when a factor appears twice.

To find zeros from factors, set each factor to zero. For

f(x) = (x - 1)(x - 2)^2

, the zeros are

x = 1

and

x = 2

. Even though

(x - 2)

appears twice,

x = 2

counts as one distinct zero (a double zero).

This is the factor idea in reverse: if

(x - a)

is a factor of a polynomial, then

x = a

is a zero, because substituting

a

makes that factor

0

. So if

3x + 4

is a factor, set

3x + 4 = 0

to get the zero

x = -\frac{4}{3}

Worked examples

Find the zeros of

f(x) = x(x + 5)

.
Set each factor to zero:

x = 0

x + 5 = 0

.
So the zeros are

x = 0

and

x = -5

How many distinct zeros does

g(m) = m^2(2m - 1)(m + 2)^3(m - 2)^4

have?
Set each distinct factor to zero:

m = 0

m = \frac{1}{2}

m = -2

m = 2

.
Repeated factors still give one value each, so there are

4

distinct zeros.

Find the zeros of

y = x^3 - 3x^2 - 4x + 12

.
Factor by grouping:

x^2(x - 3) - 4(x - 3) = (x^2 - 4)(x - 3) = (x - 2)(x + 2)(x - 3)

.
The zeros are

x = 2

x = -2

, and

x = 3

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Finding zeros of a function

Worked examples

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