Given: #(7x^3 + 40x^2 + 22x - 35)/(x+1)#
The Remainder Theorem states that when you divide a polynomial #f(x)# by a linear factor #(x-a)#, you will have a quotient function #q(x)# and a remainder.
The remainder #= f(a)#. This remainder can be found using long division, synthetic division or the Remainder Theorem.
Long Division:
#" "ul(" "7x^2 + 33x - 11)larr# quotient function
#x + 1| 7x^3 + 40x^2 + 22x - 35#
#" "ul(7x^3 + 7x^2)#
#" "33x^2 + 22x#
#" "ul(33x^2 + 33x)#
#" "-11x - 35#
#" "ul(-11x - 11)#
#" "-24 larr#remainder
Remainder Theorem:
linear factor: #(x-a) = (x +1) = (x - -1)#.
This means #a = -1#
#f(x) = 7x^3 + 40x^2 + 22x - 35#
#f(-1) = 7(-1)^3 + 40(-1)^2 + 22(-1) - 35 = -24#
Remainder #= -24#
