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?

1 Answer
Sep 23, 2015

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.)