The Correct Answer and Explanation is:
The correct answer is 3(5a – 9).
The two missing terms for the top of the area model are 5a and -9.
Explanation
The area model is a visual tool for understanding factoring, which is closely related to the distributive property. In this problem, the expression to be factored is 15a – 27. The total area of the combined rectangle represents this expression. The height and the total length of the rectangle represent its factors.
The model shows that the height of the entire rectangle is 3. This height represents the greatest common factor (GCF) of the terms 15a and -27. The GCF is the largest number that divides into both 15 and 27, which is indeed 3. This is our first factor.
Our goal is to find the length of the top side of the rectangle, which will be our second factor. The top length is composed of two separate parts, one for each smaller rectangle.
- Finding the First Term (Orange Rectangle):
The area of the first rectangle is given as 15a, and its height is 3. Since the area of a rectangle is calculated as height times length (Area = Height × Length), we can find the missing length by dividing the area by the height.
Length = Area / Height = 15a / 3 = 5a.
So, the first missing term at the top is 5a. - Finding the Second Term (Pink Rectangle):
The area of the second rectangle corresponds to the second term of our original expression, which is -27. Its height is also 3. We find its length the same way.
Length = Area / Height = -27 / 3 = -9.
So, the second missing term at the top is -9.
Now that we have both parts of the top length (5a and -9), we can write the complete second factor by combining them: (5a – 9).
To write the final expression in factored form, we multiply the height by the total length: 3 × (5a – 9), which is written as 3(5a – 9).
