Question

factor -7y + y² - 30
the factored expression is \square

Explanation:

Step1: Rearrange the terms

First, we rearrange the polynomial \(-7y + y^2 - 30\) in standard quadratic form \(y^2 - 7y - 30\). This is because the standard form of a quadratic is \(ax^2+bx + c\), which makes factoring easier. So we have \(y^2-7y - 30\).

Step2: Find two numbers

We need to find two numbers that multiply to \(a\times c=1\times(- 30)=-30\) and add up to \(b = - 7\). Let's list the factor pairs of \(-30\):

• \(1\) and \(-30\): \(1+(-30)=-29\)
• \(2\) and \(-15\): \(2+(-15)=-13\)
• \(3\) and \(-10\): \(3+(-10)=-7\)

So the two numbers are \(3\) and \(-10\).

Step3: Split the middle term

We split the middle term \(-7y\) using the two numbers we found. So \(y^2-7y - 30=y^2+3y-10y - 30\).

Step4: Factor by grouping

Group the first two terms and the last two terms:
\((y^2 + 3y)+(-10y - 30)\)
Factor out the greatest common factor (GCF) from each group. From the first group, the GCF is \(y\), so \(y(y + 3)\). From the second group, the GCF is \(-10\), so \(-10(y + 3)\). Now we have \(y(y + 3)-10(y + 3)\).

Step5: Factor out the common binomial factor

The common binomial factor is \((y + 3)\), so we factor that out: \((y + 3)(y - 10)\).

Answer:

\((y - 10)(y + 3)\)