Question

Create your own division problem and show how to use the Remainder Theorem to determine whether a binomial is a factor of a polynomial function. (Please add step by step)

Answer 1

To determine whether a binomial is a factor of a polynomial function, use the Remainder Theorem. Perform synthetic division to check if the remainder is 0. If it is, the binomial is a factor.

Step-by-step explanation:

To determine whether a binomial is a factor of a polynomial function, we can use the Remainder Theorem. The Remainder Theorem states that if the division of the polynomial function by the binomial results in a remainder of 0, then the binomial is a factor of the polynomial.

Let's create an example:

If we have a polynomial function f(x) = x^3 - 2x^2 + 4x - 8 and we want to check if (x - 2) is a factor, we can use synthetic division. Set up the division as follows:

2 | 1 -2 4 -8

Perform synthetic division:

1. Bring down the first coefficient, which is 1
2. Multiply the divisor by the current quotient term and write the result below the next coefficient (2 * 1 = 2)
3. Add the result to the next coefficient (-2 + 2 = 0)
4. Multiply the divisor by the new quotient term and write the result below the next coefficient (2 * 0 = 0)
5. Add the result to the next coefficient (4 + 0 = 4)
6. Multiply the divisor by the new quotient term and write the result below the next coefficient (2 * 4 = 8)
7. Add the result to the next coefficient (-8 + 8 = 0)

We get a remainder of 0, which means that (x - 2) is a factor of the polynomial function.