Question

How do you use the remainder theorem to evaluate `f(x)=x^3+x^2-5x-6` at x=2?

Answer 1

`f(2) = -4`

Explanation:

We use the remainder theorem to establish what the remainder is when we divide a polynomial function by a linear factor.

Here we are given a function and asked to evaluate the value of the function at a given value, so we can just evaluate it:

We have:

`f(x) =x^3+x^2-5x-6`

And so, when `x=2` we have:

`f(2) =8+4-10-6 = -4`

We can also use the remainder theorem to establish a value of `f(a)`. as the remainder theorem tells us that is we divide `f(x)` by a linear factor `(x-a)` the remainder is `f(a)`.

So, let us divide the given polynomial, `f(x)` by `x-2` using algebraic long division, yielding a remainder of `f(2)`

`{: ( , , , , x^2 , + , 3x , +, 1 , ), ( , ,"----", "----", "----", "----", "----", "----", "----" , ), (x-2, ")" , x^3, +,x^2, -,5x,-,6, ), ( , , x^3, -, 2x^2, , , , , - ), ( , , , ,"----", "----", "----", "----", "----", ), ( , , , , 3x^2, -, 5x, -, 6, ), ( , , , , 3x^2, -, 6x, , , - ), ( , , , , , ,"----", "----", "----" , ), ( , , , , , , x, -, 6, ), ( , , , , , , x, -, 2, -), ( , , , , , , , "----", "----" , ), ( , , , , , , , -, 4, ) :}`

And we have remainder `-4` indicating `f(2)=-4` , as above.