Question

Task 5—Polynomial Division and the Remainder Theorem

Create a quadratic polynomial function f(x) and a linear binomial in the form (x − a).



Part 1. Show all work using long division to divide your polynomial by the binomial.



Part 2. Show all work to evaluate f(a) using the function you created.



Part 3. Use complete sentences to explain how the remainder theorem is used to determine whether your linear binomial is a factor of your polynomial function

Answer 1

See Explanation

Explanation:

Let our quadratic polynomial function
`f(x)=x^2-7x+6`

Let our linear binomial in the form (x − a)=x-1

Part 1: We use long division to divide the polynomial by the binomial.

`\left|\begin{array}c&x-6\\-----&-----\\x-1&x^2-7x+6\\Subtract&-(x^2-x)\\&------\\&-6x+6\\Subtract&-6x+6\\&------\\&0\end{array}\right|`

Therefore:
`(x^2-7x+6)/(x-1)=x-6`

Part 2: In our chosen linear monomial x-1, a=1

`f(a)=f(1)\\f(1)=1^2-7(1)+6=1-7+6=0\\f(a)=0`

Part 3:

To determine whether a linear binomial is a factor of a polynomial function using the remainder theorem

• Given a linear binomial x-a
• Substitute a into the polynomial function f(x)
• If the value of f(a)=0, the linear binomial is a factor. However, if f(a) is not equal to zero, then by the remainder theorem, x-a is not a factor of f(x)