Factor -6 out of 18z – 15.
The Correct Answer and Explanation is:
Factoring –6 out of the Expression:
Given expression:
18z – 15
We are asked to factor –6 out of this expression.
Step-by-step Factoring:
We factor –6 out of both terms:18z−15=−6(−3z+156)18z – 15 = -6(-3z + \frac{15}{6})18z−15=−6(−3z+615)
But we can simplify further. First, divide each term by –6:18z−6=−3zand−15−6=2.5\frac{18z}{-6} = -3z \quad \text{and} \quad \frac{-15}{-6} = 2.5−618z=−3zand−6−15=2.5
But the result includes a decimal (2.5), which indicates we didn’t get a fully factored expression with integers. Let’s take another approach and look for how to factor –6 properly so the remaining expression contains whole numbers.
Better approach:
Instead of directly factoring –6, let’s express the original expression as a product of –6 and another expression.
We can write:18z−15=−6(−3z+2.5)18z – 15 = -6(-3z + 2.5)18z−15=−6(−3z+2.5)
But as shown, this results in a non-integer expression. To avoid decimals, let’s convert 2.5 to a fraction:18z−15=−6(−3z+52)18z – 15 = -6\left(-3z + \frac{5}{2}\right)18z−15=−6(−3z+25)
So, this is the correct factored form when factoring –6:18z−15=−6(−3z+52)\boxed{18z – 15 = -6\left(-3z + \frac{5}{2}\right)}18z−15=−6(−3z+25)
However, many prefer factored forms with integer coefficients. So instead, if we factor out the greatest common factor (GCF) instead of specifically –6, we get:18z−15=3(6z−5)18z – 15 = 3(6z – 5)18z−15=3(6z−5)
But since the question specifically asks to factor –6, we must accept fractional or decimal coefficients in the factored expression.
Explanation
Factoring is the process of rewriting an expression as a product of its factors. In this case, we are given the linear expression 18z – 15, and we are asked to factor –6 out of it. Factoring out a number means expressing the entire expression as that number multiplied by another simplified expression.
We begin by dividing each term in the original expression by –6. Dividing 18z by –6 gives –3z. Dividing –15 by –6 gives 2.5. Therefore, the expression becomes:18z−15=−6(−3z+2.5)18z – 15 = -6(-3z + 2.5)18z−15=−6(−3z+2.5)
This is mathematically correct but includes a decimal, which is often less preferred in algebra. Alternatively, we can write 2.5 as the fraction 5/2:18z−15=−6(−3z+52)18z – 15 = -6\left(-3z + \frac{5}{2}\right)18z−15=−6(−3z+25)
This shows that factoring –6 from the expression results in another expression containing a fraction. It is important to note that factoring a negative number can cause signs to change inside the parentheses. For example, 18z (positive) divided by –6 becomes –3z. Similarly, –15 divided by –6 becomes +2.5 because dividing two negative numbers yields a positive result.
In summary, factoring –6 out of 18z – 15 results in:−6(−3z+52)\boxed{-6\left(-3z + \frac{5}{2}\right)}−6(−3z+25)
This is correct and valid, though not the most simplified in terms of integer coefficients. However, because the question specifies factoring out –6, we must use it even if it introduces fractions.
