How to Find Zeros of a Polynomial
The zeros of a polynomial are the input values that make the polynomial equal to zero. But finding zeros, also called finding roots or x-intercepts, is one of the most important skills in algebra because it helps you solve equations, analyze graphs, simplify rational expressions, and understand the behavior of functions. If you have a polynomial such as (f(x)=x^2-5x+6), its zeros are the values of (x) for which (f(x)=0). In this case, the zeros are (x=2) and (x=3), because both values make the expression equal to zero.
What Are Zeros of a Polynomial?
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. Take this: if
[ f(x)=x^2-4, ]
then the zeros are found by solving
[ x^2-4=0. ]
Since
[ x^2-4=(x-2)(x+2), ]
the zeros are
[ x=2 \quad \text{and} \quad x=-2. ]
On a graph, the zeros of a polynomial are the points where the graph crosses or touches the x-axis. Consider this: these are also called x-intercepts. To give you an idea, the graph of (f(x)=x^2-4) crosses the x-axis at ((-2,0)) and ((2,0)) It's one of those things that adds up. But it adds up..
Finding zeros is useful because many real-world problems involve finding when a quantity becomes zero, such as profit, height, area, cost, or motion.
The Basic Idea: Set the Polynomial Equal to Zero
The first step in finding zeros is always the same:
[ f(x)=0. ]
If the polynomial is already factored, you can use the Zero Product Property. This property says that if
[ ab=0, ]
then either
[ a=0 ]
or
[ b=0. ]
For example:
[ (x-3)(x+5)=0. ]
This means:
[ x-3=0 \quad \text{or} \quad x+5=0. ]
Solving each equation gives:
[ x=3 \quad \text{or} \quad x=-5. ]
So the zeros are (3) and (-5) And that's really what it comes down to. Turns out it matters..
Step-by-Step Process for Finding Zeros
Here is a general process you can follow:
-
Write the polynomial equation in standard form.
This means arranging terms from highest degree to lowest degree And that's really what it comes down to.. -
Move everything to one side of the equation.
To find zeros, you need the equation in the form:[ f(x)=0. ]
-
Factor the polynomial if possible.
Look for common factors, special patterns, trinomial factoring, grouping, or known formulas. -
Use the Zero Product Property.
Set each factor equal to zero. -
Solve each resulting equation.
-
Check your answers.
Substitute each zero back into the original polynomial to make sure it equals zero.
Finding Zeros by Factoring
Factoring is often the easiest method when the polynomial can be broken into simpler factors.
Example 1: Linear Polynomial
Find the zero of:
[ f(x)=4x-12. ]
Set the polynomial equal to zero:
[ 4x-12=0. ]
Add 12 to both sides:
[ 4x=12. ]
Divide by 4:
[ x=3. ]
So the zero is:
[ x=3. ]
Example 2: Quadratic Polynomial
Find the zeros of:
[ f(x)=x^2+7x+12. ]
Set the polynomial equal to zero:
[ x^2+7x+12=0. ]
Factor the quadratic:
[ (x+3)(x+4)=0. ]
Set each factor equal to zero:
[ x+3=0 \quad \text{or} \quad x+4=0. ]
Solve:
[ x=-3 \quad \text{or} \quad x=-4. ]
So the zeros are:
[ x=-3 \quad \text{and} \quad x=-4. ]
Example 3: Factoring Out a Common Factor
Find the zeros of:
[ f(x)=2x^3-8x. ]
First, factor out the greatest common factor:
[ 2x(x^2-4)=0. ]
Now factor the difference of squares:
[ 2x(x-2)(x+2)=0. ]
Set each factor equal to zero:
[ 2x=0, \quad x-2=0, \quad x+2=0. ]
Solve:
[ x=0, \quad x=2, \quad x=-2. ]
So the zeros are:
[ x=0, \quad x=2, \quad x=-2. ]
Factoring by Grouping
Some polynomials do not have a common factor, but their terms can be grouped The details matter here. Turns out it matters..
Example 4: Polynomial with Four Terms
Find the zeros of:
[ f(x)=x^3+3x^2+2x+6. ]
Group the terms:
[ (x^3+3x^2)+(2x+6)=0. ]
Factor each group:
[ x^2(x+3)+2(x+3)=0. ]
Now factor out the common binomial:
[ (x+3)(x^2+2)=0. ]
Set each factor equal to zero:
[ x+3=0 \quad \text{or} \quad x^2+2=0. ]
Solve:
[ x=-3. ]
For the second factor:
[ x^2+2=0 ]
[ x^2=-2. ]
This has no real solutions, but it does have complex solutions:
[ x=\pm i\sqrt{2}. ]
So the real zero is:
[ x=-3, ]
and the complex zeros are:
[ x=i\sqrt{2}, \quad x=-i\sqrt{2}. ]
Using Special Factoring Patterns
Some polynomials can be factored using special patterns. Knowing these patterns makes finding zeros much faster Worth knowing..
Difference of Squares
[ a^2-b^2=(a-b)(a+b) ]
Example:
[ x^2-25=0 ]
[ (x-5)(x+5)=0 ]
[ x=5 \quad \text{or} \quad x=-5 ]
Difference of Cubes
[ a^3-b^3=(a-b)(a^2+ab+b^2) ]
Example:
[ x^3-8=0 ]
[ x^3-2^3=0 ]
[ (x-2)(x^2+2x+4)=0 ]
One real zero is:
[ x=2. ]
