The remainder theorem can be used to evaluate polynomials for a specific value of x as well as determine if the value is a zero of the function.
The following is the remainder theorem:
If a polynomial f(x) is divided by x - k, then the remainder is r = f(k)
Sample Math Problems
Question
Evaluate the function for f(4) when f(x) = 4x³ - 5x² + 7x - 20
Solution
One way to evaluate this is to substitute the 4 into the function. However, the remainder theorem makes it quicker to solve by hand using synthetic division. The final answer is the remainder and is also the value for when that number is evaluated into a function.
Question
Find the remainder when f(x) = 2x² - 3x + 10 is divided by (x-2)
Solution
We can use long division to determine the remainder, or we can use the remainder theorem.
- Start by identifying “k” as defined above. If you are dividing by (x - 2), then k = 2
- Evaluate the function at 2
f(2) = 2(2)² - 3(2) + 10
= 2(4) - 6 + 10
= 8 - 6 + 10
= 12
Therefore, when f(x) = 2x²-3x+10 is divided by (x-2), the remainder will be 12.
Question
Determine if x = 1 is a zero of f(x) = x³- x² + x -1
Solution
To be a zero of a function, there are no remainders when divided. Additionally, when we substitute in the value, the function will evaluate to zero, or in other words have no remainder. Let’s take a look at it in either way:
Way #1 - Synthetic Division
Way #2 - Evaluating the function
f(1) = (1)³ - (1)² + 1 - 1
= 1 - 1 + 1 - 1
= 0
Since f(1) = 0, you know there is no remainder. If there is no remainder, then the value is considered a zero of the function.
