Multimedia Link: For further explanation of the Zero Product Property, watch this: CK-12 Basic Algebra: Zero Product Property
- YouTube video.
Polynomials can be written in expanded form or in factored form. Expanded form means that you have sums and differences of different terms:
Notice that the degree of the polynomial is four.
The factored form of a polynomial means it is written as a product of its factors.
The factors are also polynomials, usually of lower degree. Here is the same polynomial in factored form.
Zero Product Property: The only way a product is zero is if one or both of the terms are zero.
By setting the factored form of the polynomial equal to zero and using this property, we can easily solve the original polynomial.
Solution: According to the property, for the original polynomial to equal zero, we have to set each term equal to zero and solve.
Solving Simple Polynomial Equations by Factoring
We already saw how we can use the Zero Product Property to solve polynomials in factored form. Here you will learn how to solve polynomials in expanded form. These are the steps for this process.
Step 2: Factor the polynomial completely.
Step 3: Use the zero-product rule to set each factor equal to zero.
Step 4: Solve each equation from step 3.
Step 5: Check your answers by substituting your solutions into the original equation.
Solve the following polynomial equation:
Separate each factor and set equal to zero:
Expanded form: Polynomials can be written in expanded form or in factored form. Expanded form means that you have sums and differences of different terms:
Factored form: The factored form of a polynomial means it is written as a product of its factors.
Zero Product Property: The only way a product is zero is if one or more of the terms are equal to zero:
Solve the following polynomial equation.
Rewrite: This is not necessary since the equation is in the correct form.
Set each factor equal to zero.
Check: Substitute each solution back into the original equation.
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Polynomial Equations in Factored Form (9:29)
- What is the Zero Product Property? How does this simplify solving complex polynomials?
Why can’t the Zero Product Property be used to solve the following polynomials?
Solve the following polynomial equations.