Question

factor the following trinomial. 2z² - 3z - 9 (z - ?)(□ z + 3)

Explanation:

Step1: Expand the given factored form

Let the first box be \( a \) and the second box be \( b \). So we have \((z - a)(bz + 3)=bz^{2}+3z - abz-3a\). Combine like terms: \(bz^{2}+(3 - ab)z-3a\).

Step2: Compare with the original trinomial

The original trinomial is \(2z^{2}-3z - 9\). So we can set up equations by comparing coefficients:

• For the coefficient of \(z^{2}\): \(b = 2\) (since the coefficient of \(z^{2}\) in \(2z^{2}-3z - 9\) is 2 and in \(bz^{2}+(3 - ab)z-3a\) is \(b\)).
• For the constant term: \(-3a=-9\). Solve for \(a\): Divide both sides by \(- 3\), we get \(a = 3\). We can also check the middle term: Substitute \(b = 2\) and \(a = 3\) into the coefficient of \(z\): \(3 - ab=3-2\times3=3 - 6=-3\), which matches the coefficient of \(z\) in the original trinomial.

Answer:

The first box (the question mark) is \(3\) and the second box is \(2\), so the factored form is \((z - 3)(2z + 3)\)