Explain why the following expression is equivalent: (z - 3)( z - 4) = z(z - 4) -3(z - 4)

Answer 1

One way we can see equivalence is by expanding each side.

So on the left we have (z-3)(z-4) = z² - 7z +12

On the right size we have z(z-4)-3(z-4) = z² - 4z - 3z + 12 = z² - 7z + 12

So we can see equivalence because both sides simplify to the same expression.

However another way to look at this is both sides are different ways to factor the same expression.

Let's look at the right side: z(z-4)-3(z-4)

z(z-4)-3(z-4)

The z and 3 that I bolded both share a common factor of (z-4), so lets pull it out front, now you get (z-4)(z-3) which is what we had on the left side.

The factoring (z-4) out would be like in the following:

abc - xyc = c(ab-xy)

in this case, c is a common factor of abc and xyc so you can pull it out front and multiply it by the rest of the expression. That concept is what is being done above by pulling out (z-4).