in the polynomial below what number should replace the question mark to produce a difference of squares x^2+? x-36

A difference of squares is exactly what it suggests - the difference between two perfect squares. 25 - 9, 4 - 1, x² - 25, and 125 - b² are just a few examples. Differences of squares factor very nicely, too. For any difference of squares x² - y²:

x² - y² = (x + y) (x - y)

We can see that this is true by taking the right side of the equation and distributing:

(x + y) (x - y) = (x + y) · x + (x + y) · (-y) = x² + xy - xy - y² = x² - y²

We notice in our given expression that 36 is a perfect square - namely, 6². We want the expression x² + ? x - 36 to look like x² - 6², which we can accomplish if we replace the question mark with a 0.

a^2 - b^2 is a difference of squares.

There is no middle term.

x^2 + 0x - 36 = x^2 - 36 = x^2 - 6^2

? = 0