Question

factor ( z^2 + 7z - 18 ). the factored expression is \\(\square\\).

Explanation:

Step1: Find two numbers

We need two numbers that multiply to -18 and add up to 7. Let's list the factor pairs of -18: (-1, 18), (1, -18), (-2, 9), (2, -9), (-3, 6), (3, -6). Now check which pair adds to 7: -2 + 9 = 7.

Step2: Factor the quadratic

Using the numbers -2 and 9, we can factor \( z^2 + 7z - 18 \) as \( (z - 2)(z + 9) \) (wait, no, wait: wait, the product is -18, so if the numbers are -2 and 9, then \( (z - 2)(z + 9) \) would expand to \( z^2 + 9z - 2z - 18 = z^2 + 7z - 18 \). Wait, actually, let's correct: the two numbers are -2 and 9? Wait, no, wait: let's do it properly. The quadratic is \( z^2 + bz + c \), here \( b = 7 \), \( c = -18 \). So we need two numbers that multiply to \( c = -18 \) and add to \( b = 7 \). So the numbers are 9 and -2, because 9 (-2) = -18, and 9 + (-2) = 7. So the factored form is \( (z + 9)(z - 2) \)? Wait, no: \( (z + a)(z + b) = z^2 + (a + b)z + ab \). So if \( ab = -18 \) and \( a + b = 7 \), then a and b are 9 and -2, because 9 (-2) = -18, 9 + (-2) = 7. So \( (z + 9)(z - 2) \) would be \( z^2 + 9z - 2z - 18 = z^2 + 7z - 18 \), which is correct. Wait, but actually, it's \( (z - 2)(z + 9) \) or \( (z + 9)(z - 2) \), same thing.

Answer:

\( (z - 2)(z + 9) \) (or \( (z + 9)(z - 2) \))