Factor the following trinomials completely.
$z^2-11z+30$

• A. $(z+5)(z-6)$
• B. $(z-3)(z-10)$
• C. $(z+5)(z+6)$
• D. $(z-5)(z-6)$

Answer: (D)

Verified step by step guidance

1. Identify the trinomial to factor: \(z^2 - 11z + 30\).
2. Look for two numbers that multiply to the constant term \(30\) and add up to the coefficient of the middle term, which is \(-11\).
3. List the factor pairs of \(30\): \((1, 30)\), \((2, 15)\), \((3, 10)\), and \((5, 6)\), and consider their signs to sum to \(-11\).
4. Since the middle term is negative and the constant is positive, both numbers should be negative. Check which pair of negative numbers multiply to \(30\) and add to \(-11\).
5. Write the factored form using the two numbers found: \((z - a)(z - b)\), where \(a\) and \(b\) are the numbers identified in the previous step.