Use the Polynomial Remainder Theorem to analyze factors of polynomials

The  exercise appears under the Algebra II Math Mission. This exercise practices using the PRT (Polynomial Remainder Theorem) to determine the factors of polynomials and their remainders when divided by linear expressions.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find the remainder then evaluate if it is a factor: This problem states that a particular polynomial is divided by a linear binomial and asks for the remainder. Students are expected to find the remainder, and state whether the expression is a factor of the polynomial.Use the Polynomial Remainder Theorem to analyze factors of polynomials.PNG
 * 2) Find the remainder and factor: This problem provides a polynomial and a binomial that needs to be a factor of the polynomial. The student is expected to find the correct value for the remainder and factor that will work.Use the Polynomial Remainder Theorem to analyze factors of polynomials2.PNG

Strategies
Knowledge of the remainder theorem and other polynomials theorems are encouraged to ensure success on this exercise.
 * 1) The remainder theorem states that the numerical remainder when one divides $$p(x)$$ by $$x-r$$, is just $$p(r)$$.
 * 2) A corollary, the factor theorem, states that $$x-r$$ being a factor of a polynomial is equivalent to $$p(r)$$ evaluating to zero. Another motivation for calling these values zeroes.
 * 3) A divisor is a factor of the polynomial only when the remainder is zero. This is because polynomial division is basically the same as dividing integers.

Real-life Applications

 * 1) The remainder theorem provides a more efficient avenue for testing whether certain numbers are roots of polynomials.
 * 2) This theorem can increase efficiency when applying other polynomial tests, like the rational roots test.