To determine which model represents the factors of the quadratic expression [tex]\(x^2 + 9x + 8\)[/tex], we first factor the quadratic expression.
Given the expression:
[tex]\[ x^2 + 9x + 8 \][/tex]
The factors of this expression are:
[tex]\[ (x + 1)(x + 8) \][/tex]
Let's verify this through the following steps:
1. Identify the factors:
To factor the quadratic expression [tex]\( x^2 + 9x + 8 \)[/tex], we need to find two binomials [tex]\((x + a)(x + b)\)[/tex] such that when multiplied, they yield the original quadratic expression. Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers.
2. Expand the factors to ensure correctness:
Let's expand [tex]\((x + 1)(x + 8)\)[/tex]:
[tex]\[
(x + 1)(x + 8) = x(x + 8) + 1(x + 8)
\][/tex]
Distribute [tex]\(x\)[/tex] and [tex]\(1\)[/tex]:
[tex]\[
x^2 + 8x + 1x + 8
\][/tex]
Combine like terms:
[tex]\[
x^2 + 9x + 8
\][/tex]
3. Match it with the original:
The expanded form matches the original quadratic expression [tex]\(x^2 + 9x + 8\)[/tex], confirming that the factors are indeed correct.
Given the factors [tex]\((x + 1)(x + 8)\)[/tex], we can check if the given model represents this factorization.
4. Interpret the model:
The provided model is a table. Let's analyze each part of the table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline & $+x$ & + & + & + & + \\
\hline $+x$ & \begin{tabular}{l}
$+x^2$
\end{tabular} & $+x$ & $+x$ & $+x$ & $+x$ \\
\hline + & $+x$ & + & + & + & + \\
\hline + & $+x$ & + & + & + & + \\
\hline
\end{tabular}
\][/tex]
Based on this table:
- The elements in the first column and first row (excluding the first cell) '+x', suggest that these are related to the linear terms (coefficients of [tex]\(x\)[/tex]) from the factors.
- The top row, when combined with the first column elements, implies multiplicative interactions or cross products that should yield the quadratic expression.
Since the box suggests the components of each factor, if the table is carefully drawn to represent the multiplication of [tex]\((x + 1)\)[/tex] and [tex]\((x + 8)\)[/tex], the terms inside should collectively sum up to form [tex]\(x^2 + 9x + 8\)[/tex].
Overall, the model correctly represents the factorized form of the quadratic expression [tex]\(x^2 + 9x + 8\)[/tex].
