2. Polynomials, Linear Factors, and Zeros - 5. Polynomials and Polynomial Functions

We want to write a polynomial function in standard form with the given zeros. To do so, we will use the Factor Theorem to write the factored form. We will then simplify it by applying the Distributive Property. Let's first recall the Factor Theorem.

Factor Theorem
The expression x-a is a factor of a polynomial if and only if the value a is a zero of the related polynomial function.

We know that - 1, 1, and 0 are zeros of our function. Therefore, we can write our polynomial function as the product of three factors. f(x) = ( x-( - 1) ) ( x- 1) ( x- 0) ⇕ f(x)=(x+1)(x-1)x Finally, we can apply the Distributive Property to express the function in standard form.

f(x)=(x+1)(x-1)x

Distr

Distribute x

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

▼

Simplify right-hand side

Distr

Distribute (x^2-x)

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

Distr

Distribute x

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

RemovePar

Remove parentheses

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

AddTerms

Add terms

f(x)=x^3-x

Checking Our Answer

We can check our answer by substituting the given zeros for x. If the result is f(x)=0, it means that the given numbers are actually zeros of the function and our answer is correct. Let's start by checking - 1.

f(x)=x^3-x

Substitute

x= - 1

f( - 1)=( - 1)^3-( - 1)

▼

Evaluate right-hand side

CalcPow

Calculate power

f(- 1)=- 1-(- 1)

NegNeg

- (- a)=a

f(- 1)=- 1+1