amyduran7
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Complete the factorization of [tex]\(x^2 + 27x + 162\)[/tex] represented by the model.

[tex]\[
\begin{tabular}{|c|c|c|}
\cline{2-3}
& $x$ & \\
\hline
$x$ & $x^2$ & \\
\hline
18 & & 162 \\
\hline
\end{tabular}
\][/tex]

A. 27, [tex]\(9x, 18x\)[/tex]

B. 9, [tex]\(9x, 18x\)[/tex]

C. 27, [tex]\(9x, 27x\)[/tex]

D. 9, [tex]\(9x, 27x\)[/tex]



Answer :

GinnyAnswer
To complete the factorization of the polynomial [tex]\(x^2 + 27x + 162\)[/tex], let's analyze the steps involved:

1. Identify two numbers that both multiply to the constant term (162) and add up to the coefficient of the linear term (27).

2. The two numbers we've identified are 9 and 18.

3. Using these numbers, we can split the middle term [tex]\(27x\)[/tex] into two terms: [tex]\(9x\)[/tex] and [tex]\(18x\)[/tex]. This allows us to rewrite the expression as [tex]\(x^2 + 9x + 18x + 162\)[/tex].

4. Next, we group the terms as follows:
[tex]\[ x^2 + 9x + 18x + 162 = (x^2 + 9x) + (18x + 162) \][/tex]

5. Factor out the greatest common factor in each group:
[tex]\[ = x(x + 9) + 18(x + 9) \][/tex]

6. Notice that [tex]\((x + 9)\)[/tex] is a common factor:
[tex]\[ = (x + 9)(x + 18) \][/tex]

Therefore, the correct terms that complete the factorization [tex]\(x^2 + 27x + 162\)[/tex] are:

[tex]\[ \boxed{9, 9x, 18x} \][/tex]

So the correct answer from the given options is:

[tex]\[ 9, 9x, 18x \][/tex]

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