Factoring (Additional Techniques)

Question

Consider the expression x^2 - c. Suppose this expression can be factored as

x^2 - c = (x-3) (x+k)

What are the values of c and k?

Answer 1

Since x² - c = (x-3)(x+k), it must be true that their difference is 0 for all values of x. Therefore,

x² - c - (x-3)(x+k) = x² - c - (x² +(k - 3)x - 3k) = (3-k)x +3k - c = 0.

Letting f(x) = (3-k)x + 3k - c, since we assume f(x) = 0 for all x, we can evaluate at x = 0 to see that

f(0) = 3k - c = 0, which implies that c = 3k.

Therefore we can write f(x) = (3-k)x = 0 for all x, and by the zero product property of the real numbers, we must have 3 - k = 0, since this equality must hold for nonzero x. We conclude that k = 3, which gives us that c = 3(3) = 9.

Answer 2

x² - c could be factored into two binomials if it were a difference of squares

c = 9, thus k = 3