1. Explain why the expression 133x^3-125y^3 is called a difference of two cubes. State the general expression for the factored form of a difference of two cubes. Factor the given expression.
2. The polynomial -x3-vx2+2x+w has a remainder of 4 when divided by x+2 and a remainder of 119 when divided by x-3. What are the values of v and w? (v & w eR)
3. The volume of a box is V(x)=x^3-15x^2+66x-80. Find expressions for the dimensions of the box in terms x.
4. a). Calculate (x^5+9x^4-3x^3+5x+8) / (x^2+3x+1) using long division.
b). Write a division statement.
5. Calculate (-2x^4 +x^3-8x^2+11) / (x-6) using synthetic division.
______________________________________________________________
6. A biopsy was performed by Dr. Andersen and a 200mm by 200 mm image section of the tissue was placed on top of a Cartesian plane with its center at the origin. The darker, shaded areas found on the tissue sample satisfy the rational inequality. You are given the value of ‘a’ y=2x^2-3/x+3
a. Sketch (by hand) a graph of the function given.
b. Shade the region(s) of the graph that satisfies the inequality to
reveal the light and dark regions of the biopsy noted by Dr.
Andersen.
7. Determine the domain, intercepts, asymptotes (if any), and positive/negative intervals. Use these characteristics to sketch the graph of the function on grid paper using pencil and ruler. Then describe where the function is increasing or decreasing. Show all work. Equation: y=2x^2-3/x+3
7. Find:
> Domain
> Intercepts
> Asymptote(s)
> Negative Intervals
> Positive Intervals
> Increasing Intervals
> Decreasing Intervals
> End Behaviours
I generally eschew homework dumps, but here's a few ideas
#1. surely that is obvious
#2. Use the Remainder Theorem. That is, if f(x) = -x^3-vx^2+2x+w
then f(-2) = 4 and f(3) = 119
That gives you two equations in v and w to solve
#3. x^3-15x^2+66x-80 has possible rational roots of 1,2,4,5,16,20,40,80
So look for the low-hanging fruit first. A little synthetic division yields
(x-2)(x^2-13x+40)
and that's easy to factor
#4. There are several handy online long-division calculators. But surely you can do this simple division by hand.
#5. see #4
#6. See #4, but for graphing web sites.
#7. See #6.