Andrew M. answered • 01/27/13
Math Teacher | Instructional Designer
Something many people forget or are never told is that factors are objects being multiplied or divided while terms are objects being added or subtracted.
In this case, we're looking for a multiplication. Remember that, in math, two objects written against each other are multiplied. (i.e. 2x is 2 multiplied by x)
In this case, (x-3)(x-7) means (x-3) is being multiplied by (x-7). This makes each of those two objects factors. (They are referred to as polynomials because there are multiple "terms" inside of them. (x-3) is made up of x and -3, though that's outside the scope of this problem.)
They're linear because, if you took each individual factor and used it for an equation (i.e. y = x-7), it would produce a line. (there's more to the actual definition, but it boils down to a constant slope or ratio between x and y, which is what any line has)
Thus, the linear factors of that equation are (x-3), and (x-7).
To multiply the factors together, we use distribution. If you had something like this: 3(x+4), you would distribute the 3 across the x and the 4 through multiplication (the operation implied by placing the 3 against the parenthesis. Thus, it would expand into 3x + 12. This same principle would be used in your expression, but with some extra steps.
First, choose which factor you're going to distribute first. For an example, I'll distribute the (x-3) first.
Rather than breaking both factors apart at once, just do one at a time. For instance:
(x-3)(x-7) = x(x-3) - 7(x-3)
As you can see, the second factor is made up of x and -7. I distributed the (x-3) across each of those terms, so the x and the -7 each have one. Now, we just apply the same process as before to each of them.
x(x-3) = x2 -3x, -7(x-3) = -7x +21
So, the next part is:
x2 - 3x -7x + 21
Then, we just simplify by combining like terms. The -3x and -7x combine to make -10x
y = x2 - 10x + 21
