Remainder theorem of polynomials

The Remainder theorem of polynomials exercise appears under the Algebra II Math Mission and Mathematics III Math Mission. This exercise applies the remainder theorem about polynomials to specific problems.

Types of Problems

There are three types of problems in this exercise:

1. Determine the unknown to make something a factor: This problem provides a polynomial and a binomial that needs to be a factor of the polynomial. The user is expected to find the correct value of c that will work.

   

   Determine if the linear binomial is a factor

2. Apply the remainder theorem to find the remainder: This problem states that a particular polynomial is divided by a linear binomial and asks for the remainder. The user is expected to find the remainder, and state whether the binomial expression was a factor of the polynomial.

   

   Apply remainder theorem to find remainder

3. Select whether or not it is a factor: This problem provides a polynomial and a proposed factor. The user is asked to determine whether or not the proposed factor is actually a factor or not.

   

   Select whether or not it is a factor

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.

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.
3. Knowledge of algebra is essential for higher math levels like trigonometry and calculus. Algebra also has countless applications in the real world.