Create a third degree polynomial that when divided by x + 2 has a remainder of –4.

Question

anonymous · Answer

using synthetic division

anonymous · Answer

Dividend = divisor(quotient) + remainder
we have the divisor as (x+2) and remainder -4
third degree means that maximum value of variable has to be 3
so we choose quotient as x^2 (so that when multiplied by x+2, we get x^3)
required polynomial = x^2(x+2) + (-4) = x^3 + 2x^2 -4

asnaseer · Answer

general 3rd degree polynomial can be written as:$$ax ^{3}+bx ^{2}+cx+d$$since remainder has no terms in x^2 we know "a" must be 1. so we can rewrite as:$$x ^{3}+bx ^{2}+cx+d$$in order to get x^3 we must multiply (x+2) by x^2 which gives x^3+2x^2. so "b" must be 2 as we have no x^2 terms in the remainder. this gives us:$$x ^{3}+2x ^{2}+cx+d$$since (x+2) multiplied by x^2 gave us x^3+2x^2, we can take these terms off to get the remainder:$$cx+d$$since remainder should not have any terms in "x", "c" must be 1:$$x+d$$to get a remainder of -4 when this is divided by (x+2):$$d-2=-4$$$$d=-2$$so the equation we want is:$$x ^{3}+2 x ^{2}+x-2$$