For exercises 5-6 on page 9 out of 10 on introduction to polynomials I need some help. for y=f(x)=(x-3)(x+2), does it matter what numbers we use for x, and what would a table and graph of that look like. I think i know what im doing but not sure.
When you graph y vs. x, you need to figure out how y will change as x varies over its domain.
The first values of x you want to look at are those values that make y equal to zero. For the given function, those values would be x = 3 and x = -2.
Also you want to consider what happens for x = 0. In this case you get y = f(0) = (0 - 3) ( x + 2) = -3 * 2 = -6.
So at this point you have the beginnings of a table:
x y
-2 0
0 -6
3 0.
You could plot these points and get a pretty good idea how the graph behaves.
Then considering that when x gets very large the product (x - 3) ( x + 2) also gets very large, you get a good idea of what the graph will look like.
For exercises 5-6 on page 9 out of 10 on introduction to polynomials I need some help. What is the difference in using y=f(x)=(x-3)(x+2)instead of one with three linears, like y=f(x)=(x-3)(x+2)(x+1), do we just make another column in the table or what?
For this function we also have y = 0 when x = -1. When x = 0 we still have y = -6. So our table would now look like this:
x y
-2 0
-1 0
0 -6
3 0.
If x is a large negative number, then (x - 3) ( x + 2) ( x + 1) would be the product of three large negative numbers, which would be a very large negative number.
If x is a large positive number, then (x - 3) ( x + 2) ( x + 1) would be the product of three large positive numbers, which would be a very large positive number.
So the graph starts with large negative values, passes through the x axis at the indicated points (and only at these points), then continues through large positive values.
What is an irreducible quadratic factor and how is it different from a linear factor, and how is it used?
An irreducible quadratic factor is a factor a x^2 + b x + c that can never be zero. This is equivalent to say that it can't be factored.
a x^2 + b x + c is never zero if b^2 - 4 a c is negative.
For example the polynomial x^3 + x^2 + x - 3 can be factored into the form
(x - 1) ( x^2 + 2x + 3).
This expression is zero when either x - 1 = 0 or x^2 + 2 x + 3 = 0.
x - 1 = 0 when x = 1.
x^2 + 2 x + 3 is never equal to zero, since b^2 - 4 a c = 2^2 - 4 * 1 * 3 = -8.
So the only way x^3 + x^2 + x - 3 can be zero is if x = 1.
The graph therefore goes through the x axis only at x = 1.