Answered

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The polynomial [tex]$24 x^3-54 x^2+44 x-99$[/tex] is factored by grouping.

[tex]\[
\begin{array}{l}
24 x^3-54 x^2+44 x-99 \\
24 x^3+44 x-54 x^2-99 \\
4 x(\ldots)-9(\ldots \ldots)
\end{array}
\][/tex]

What is the common factor that is missing from both sets of parentheses?

A. [tex]6 x + 11[/tex]
B. [tex]6 x - 11[/tex]
C. [tex]6 x^2 + 11[/tex]
D. [tex]6 x^2 - 11[/tex]

Sagot :

GinnyAnswer
To factor the polynomial [tex]\(24x^3 - 54x^2 + 44x - 99\)[/tex] by grouping, let's break down the steps:

1. Group the terms appropriately:
[tex]\[ 24x^3 - 54x^2 + 44x - 99 \][/tex]
Group as:
[tex]\[ (24x^3 - 54x^2) + (44x - 99) \][/tex]

2. Factor out the greatest common factor (GCF) from each group:
- For the first group [tex]\(24x^3 - 54x^2\)[/tex], the GCF is [tex]\(6x^2\)[/tex]:
[tex]\[ 24x^3 - 54x^2 = 6x^2(4x - 9) \][/tex]
- For the second group [tex]\(44x - 99\)[/tex], the GCF is [tex]\(11\)[/tex]:
[tex]\[ 44x - 99 = 11(4x - 9) \][/tex]

3. Notice that after factoring out the GCF from each group, the binomial factors [tex]\(4x - 9\)[/tex] are common:
[tex]\[ 24x^3 - 54x^2 + 44x - 99 = 6x^2(4x - 9) + 11(4x - 9) \][/tex]

4. Factor out the common binomial factor [tex]\((4x - 9)\)[/tex]:
[tex]\[ 6x^2(4x - 9) + 11(4x - 9) = (4x - 9)(6x^2 + 11) \][/tex]

Therefore, the common factor that is missing from both sets of parentheses is:
[tex]\[ 6x^2 + 11 \][/tex]

So, the correct answer is:
```
6 x^2 + 11
```
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