Question

How do you use synthetic division and the Remainder Theorem to find P(a) `P(x) = 2x^3 - x^2 + 10x + 5` ;a = 1/2?

Answer 1

Divide `P(x) = 2x^3 - x^2 + 10x + 5` by `x-a` for `a = 1/2`

Here is the synthetic division:
(You could use long division to get the number instead. But your teacher/grader may want to choose the type of division to check your knowledge of it.)

`1/2|2" "-1color(white)(XX)10" "color(white)(X)5`
`color(white)(1)|" "color(white)(XX1)1" "color(white)(X)0color(white)(XX1)5`
`" "stackrel("—————————————)`
`color(white)(1)|2" "color(white)(XX)0" "color(white)(1)10" ""|"color(white)(1)10`

The remainder is `10`,
The remainder Theorem says that when we divide a polynomial `P(x)` by `x-a`, the remainder is `P(a)`

When we divided this `P(x)` by `x-1/2` we got remainder `10`.

So, `P(1/2) = 10`

(Division format from Ernest Z.)