6. Consider the cubic polynomial p(x) = x^3 + x^2 - 46x + 80.
(a) Using polynomial long division, write the ratio of p(x)/(x-3) in quotient-remainder form, i.e. in the form q(x) + (r)/(x-3). Evaluate p(3). How does this help you check your quotient-remainder form?
x^2+4x-34 - 22/(x-3)
x-3 | x^3+x^2-46x+80
Basically with remainder -22/(x-3) then p(3) will equal -22
See cphill's explanation/answer here for the second part:
Using EP's answer of R = -22
We should get the same answer by subbing 3 inito the polynomial....so...
(3)^3 + (3)^2 - 46(3) + 80 =
27 + 9 - 138 + 80 =
116 - 138 =
-22 ......Magic !!!!!
Wait... I don't understand by the question: How does this help you check your quotient-remainder form?
It helps because the "r" in r / [ x - 3 ] should be the same as evaluating P(a)
So...in the case EP divided the polynomial by x - a = x - 3 got a remainder of -22
Then....we can check whether this remainder is correct by evaluating P(a) = P(3)
Note that when we evaluated P(3)....we put 3 into the polynomial and got -22
So....we can be sure that this remainder is correct....!!!!!
Does that make sense, GM???