Factoring techniques and rational expressions in algebra

Factoring

Week Two Discussion Questions

How do you factor the difference of two squares?

This expression is called a difference of two squares:

The factors of a2 - b2 are:

(a + b) and (a -- b)

How do you factor the perfect square trinomial?

Both x2 and 9 are perfect squares.

Because subtraction is occurring between these squares, this expression is the difference of two squares.

x2 = x * x

The factors are (x + 3) and (x - 3).

(x + 3) (x - 3) or (x - 3) (x + 3)

How do you factor the sum and difference of two cubes?

The sum of two cubes is factored like this:

a3 + b3 = (a + b)(a2 -- ab + b2)

The difference of two cubes is factored like this:

= (a -- b)(a2 + ab + b2)

Which of these three makes the most sense to you? Explain why.

Factoring the perfect square trinomial makes the most sense to me because the calculations seem intuitive -- nothing unexpected or complex happens. Either a square exists or it does not; factoring is a relatively simple matter for squares.

Week Three Discussion Questions

Do all rational equations have a single solution? Why is that so?

Not all rational equations have a single solution.

Given, for example:

f (x) = (x2 + 2x + 1)/(x^4 + 2)

Then, x can be any real value, which will result in many possible solutions.

What constitutes a rational expression? How would you explain this concept to someone unfamiliar with it?

A rational express is the ratio of two polynomials. It is considered "rational" because one number is divided by another number:

x2 + 5 / x + 2

This operation is just like what occurs in a ratio. But note that the polynomial that you are dividing by cannot be zero.

Week Four Discussion Questions

Write a word problem involving a quadratic function. How would you explain the steps in finding the solution to someone not in this class?

Given the area of a rectangle is 560 square inches. The length is 3 more than twice the width. Find the length and the width.

L = length

W = width.

Since the length is 3 more than twice the width, then:

L = 2W + 3

The area of the rectangle is 560, so:

LW = 560

Use L = 2W +3 to solve for W:

LW = 560

(2W = 3)W = 560

2W2 + 3W = 560 (Subtract 560 from each side of the equation)

2W2 + 3W -- 560 = 0

Use the Quadratic Formula:

W = -3+/- ? 9 -- 4 (2) (-560) / 2.2 =

W = -3+/- ?4489 / 4 = -3+/- 67 / 4 = -70 / 4 Or 64 / 4 = 16

But since the width can't be negative, then:

L = 2 * 16 + 3 = 35

What is the relationship between exponents and logarithms? How would you distinguish between the two, using both a graph and a sequence?

Just like subtraction will "undo" addition, logarithms are the opposite of exponentials, and can "undo" them. Logarithms are the inverse of exponentials. Logarithms answer the question: "Which exponential produced this (the problem being solved) equation?"

For example y = bx is the equivalent of logb (y) = x

Sequentially, whatever had been the argument of the log can be the "equals" and what ever had been the "equals" can be the exponent in the exponential.

As X gets larger on the graph, the function value of f (x) increases dramatically. So the function is called an exponential function.