Questions Involving Dividing Polynomials

1) 8x^3 + 10x^2 - px - 5 is divisible by 2x+1. There is no remainder. Find the value of p.

2) When x^6 + x^4 - 2x^2 + k is divided by 1 + x^2, the remainder is 5. Find the value of k.

3) The volume of a cylindrical can is (4πx^3 + 28πx^2 + 65πx + 50π) cm^3. The can is (x +2) cm high. What is the radius?

If anyone can help me out with any of these questions I would really appreciate it! Thanks in advance! Just to clarify, "π" is pi.

Re: Questions Involving Dividing Polynomials

1.) Let $\displaystyle f(x)=8x^3+10x^2-px-5$

From the given information, we know from the remainder theorem we must have:

$\displaystyle f\left(-\frac{1}{2} \right)=0$

From this you can find $\displaystyle p$.

2.) Let $\displaystyle u=x^2$ and $\displaystyle f(u)=u^3 + u^2 - 2u + k$

By the remainder theorem and the given information, we know:

$\displaystyle f(-1)=5$

From this you can find $\displaystyle k$.

3.) The volume of the can is:

$\displaystyle V=\pi r^2h=\pi r^2(x+2)=\pi(4x^3 + 28x^2 + 65x + 50)$

From this you may determine $\displaystyle r(x)$. Check to see if the cubic has -2 as a root.

Re: Questions Involving Dividing Polynomials