Questions: Factor the trinomial completely by using any method. Remember 5 y^2 - 10 y - 40 =

Transcript text: Factor the trinomial completely by using any method. Remem \[ 5 y^{2}-10 y-40= \]

Solution

Solution Steps

Solution Approach

To factor the trinomial \(5y^2 - 10y - 40\), we can use the method of factoring by grouping. First, we look for two numbers that multiply to the product of the leading coefficient (5) and the constant term (-40), which is -200, and add up to the middle coefficient (-10). Once we find these numbers, we can split the middle term and factor by grouping.

Step 1: Identify the Trinomial

We are given the trinomial \(5y^2 - 10y - 40\) and need to factor it completely.

Step 2: Find Two Numbers for Factoring by Grouping

To factor by grouping, we need two numbers that multiply to the product of the leading coefficient and the constant term, \(5 \times (-40) = -200\), and add up to the middle coefficient, \(-10\).

Step 3: Split the Middle Term

The numbers \(-20\) and \(10\) satisfy these conditions because \(-20 \times 10 = -200\) and \(-20 + 10 = -10\). We can rewrite the trinomial as: \[ 5y^2 - 20y + 10y - 40 \]

Step 4: Factor by Grouping

Group the terms: \[ (5y^2 - 20y) + (10y - 40) \]

Factor out the greatest common factor from each group: \[ 5y(y - 4) + 10(y - 4) \]

Step 5: Factor Out the Common Binomial

Since \((y - 4)\) is common in both groups, factor it out: \[ (5y + 10)(y - 4) \]

Step 6: Simplify the Expression

Factor out the greatest common factor from \(5y + 10\): \[ 5(y + 2)(y - 4) \]