91Ó°ÊÓ

Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend $$ 2 x^{4}-3 x^{3}+x+1 \text { divided by } 2 x^{2}+x+1 $$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(4 x^{-1} y^{1 / 3}\right)^{3 / 2}}{(x y)^{3 / 2}} $$

Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend $$ 3 x^{4}-x^{3}+x-2 \text { divided by } 3 x^{2}+x+1 $$

Use the Distributive Property to remove the parentheses. $$ (x-4)(x-2) $$

Explain to a fellow student when you would use the LCM method to add two rational expressions. Give two examples. of adding two rational expressions: one in which you use the \(\mathrm{LCM}\) and the other in which you do not.

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 8 x^{2}+6 x-2 $$

Find the value of each expression if \(x=2\) and \(y=-1\). \(x^{2} y^{2}\)

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{x}{(1+x)^{1 / 2}}+2(1+x)^{1 / 2} \quad x>-1$$

Find \(k\) if \(3 x(x-5 k)=3 x^{2}-60 x\).

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{4}-81 $$