Polynomial Division: Finding Q(X) and R(X)

[stefanB]

Hello, I face this problem:

X^3 + X - 71 = (X^2 + 4X + 16)Q(X) + R(X), where Q and R are polynomials. Decide which they are.

I got that Q(X) = (X + 1/4) and that R(X) = - 75, but apparently it is wrong. I am stuck and don't know what to do.

Thanks in advance.

 

[EnumaElish]

I think R(x) should include a term with x or perhaps x^2 to cancel out the redundant x's and perhaps the x^2's created by (X^2 + 4X + 16)Q(x).

 

[chyo]

You let Q(x) = Ax + B and R(x) = Cx + D.
Substitute it into the equation and you'll get (X^2 + 4X + 16)(Ax + B) + (Cx + D).
The next step is for you to expand the above expression as per normal. Then you group the terms according to the degree of x and then equate the coefficients accordingly with X^3 + X - 71.
As a check, you should get A=1.

 

[Dick]

You could just use polynomial division to divide x^3+x-71 by x^2+4x+16 to get a quotient and remainder, which would be Q(x) and R(x) and eliminate this guesswork.