Zero theorem for the roots of a polynomial function
The zero theorem lets you calculate the roots of a polynomial function
In this lesson we’ll learn how to use the zero theorem to calculate the roots of a factored polynomial.
We can use the zero theorem to find the roots of a polynomial function once it’s been factored. When a polynomial is factored, the zero theorem tells us that, in order for the left-hand side to be equal to ???0???, one or both of the factors must be ???0???.
For example, if you have the factored polynomial
???(2x+5)(x-3)=0???
then according to the zero theorem we can set each part equal to ???0??? to find any solutions to the equation.
We can say,
???2x+5=0???
???2x+5-5=0-5???
???2x=-5???
???\frac{1}{2}\cdot 2x = \frac{1}{2} \cdot -5???
???x=\frac{-5}{2}???
???x=-\frac{5}{2}???
and
???x-3=0???
???x-3+3=0+3???
???x=3???
Step-by-step examples using the zero theorem
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How to use the zero theorem to find the solutions for a polynomial function
Example
Find the solutions of the equation.
???y=x^2-13x+36???
The roots of the equation are where the ???y???-value equals ???0???.
We set up the equation
???x^2-13x+36=0???
and we’ll factor the left-hand side.
???(x-4)(x-9)=0???
Zero theorem tells us that, in order for the left-hand side to be equal to ???0???, one or both of the factors must be ???0???. Therefore, we can say
???x-4=0???
???x-4+4=0+4???
???x=4???
and
???x-9=0???
???x-9+9=0+9???
???x=9???
The roots are ???x=4??? and ???x=9???.
Another example of finding the roots of a function
Example
Find the zeros of the function.
???f(x)=5x^2-8x+3???
Finding the zeros of a function means finding the values of ???x??? when ???f(x)??? equals ???0???.
Let’s set the function equal to ???0??? and factor.
???5x^2-8x+3=0???
???(5x-3)(x-1)=0???
Zero theorem tells us that, in order for the left-hand side to be equal to ???0???, one or both of the factors must be ???0???. Therefore, we can say
???5x-3=0???
???5x-3+3=0+3???
???5x=3???
???\frac{1}{5}\cdot 5x = \frac{1}{5} \cdot 3???
???x=\frac{3}{5}???
and
???x-1=0???
???x=1???
The zeros are ???x=3/5??? and ???x=1???.