Question

which terms complete the factorization of ( x^2 + 27x + 162 ) represented by the model?
options:

• 27, 9x, 18x
• 9, 9x, 18x
• 27, 9x, 27x
• 9, 9x, 27x

(model: a grid with cells labeled ( x ), ( x^2 ), 18, 162)

Explanation:

Step1: Analyze the area model for factoring

The quadratic is \(x^{2}+27x + 162\). In the area model (rectangle method for factoring), we know that the product of the first terms (top - left and top - middle) should be \(x^{2}\), and we already have \(x\) and \(x^{2}\), so the top - middle term (let's call it \(a\)) and \(x\) multiply to \(x^{2}\), so \(a=x\)? Wait, no, let's think about the bottom row. The bottom - left is 18, bottom - right is 162. So the term in the middle of the bottom row (let's call it \(b\)) should satisfy \(18\times b=162\), so \(b = \frac{162}{18}=9\). Also, the middle term of the quadratic is \(27x\), which is the sum of the terms in the middle column and middle row. Let's denote the top - right term as \(m\) and the middle - right term as \(n\). We know that the first row is \(x\) and \(m\), the second row is \(x\), \(x^{2}\), \(n\), the third row is 18, 9, 162. Now, the product of the first column: \(x\times18\)? No, wait, the area model for factoring \(x^{2}+bx + c=(x + p)(x + q)\) where \(p\times q=c\) and \(p + q=b\). So we need to find two numbers \(p\) and \(q\) such that \(p\times q = 162\) and \(p+q=27\). Let's factor 162: \(162=18\times9\) and \(18 + 9=27\). So the factors are \((x + 9)(x + 18)\). Now, looking at the area model: the top - right term should be 9 (because when we multiply \(x\) (top - left) and 9 (top - right), and \(x\) (second - left) and 18 (third - left)? Wait, no, let's fill the model. The first row: \(x\) and 9 (since one factor is \(x + 9\)). The second row: \(x\), \(x^{2}\), and \(18x\) (because \(x\times18 = 18x\)). The third row: 18, \(9\times18 = 162\)? Wait, no, the middle of the third row is \(9\times18\)? No, the middle cell (second row, second column) is \(x^{2}\), which is \(x\times x\). The second row, third column: \(x\times18=18x\). The first row, second column: 9 (because \(x\times9 = 9x\) and \(18\times9 = 162\)). The middle term of the quadratic \(27x=9x + 18x\). So the terms to complete the factorization are 9, \(9x\), \(18x\).

Step2: Verify the options

Let's check the options:

• Option 1: 27, 9x, 18x. The top - right term can't be 27 because \(x\times27=27x\) and \(18\times27 = 486

eq162\), so incorrect.

• Option 2: 9, 9x, 18x. Let's check: Top - right is 9, so first factor is \(x + 9\). Third - left is 18, so second factor is \(x + 18\). Then \(x\times18 = 18x\) (middle - right), \(x\times9=9x\) (middle - top), and \(18\times9 = 162\) (bottom - right). Also, \(9x+18x = 27x\), which matches the middle term. So this works.
• Option 3: 27, 9x, 27x. \(x\times27 = 27x\), \(18\times27=486

eq162\), and \(9x + 27x=36x
eq27x\), incorrect.

• Option 4: 9, 9x, 27x. \(9x+27x = 36x

eq27x\), incorrect.

Answer:

B. 9, 9x, 18x