College

Factor by grouping: [tex]x^3 - 6x^2 + 8x - 48[/tex]

Select the correct choice below and fill in any answer boxes within your choice.

A. [tex]x^3 - 6x^2 + 8x - 48 = \square[/tex]

B. The polynomial cannot be factored.

Answer :

Sure! Let's factor the polynomial [tex]\( x^3 - 6x^2 + 8x - 48 \)[/tex] by grouping.

1. Group the terms: We'll group the terms into two pairs: [tex]\( (x^3 - 6x^2) \)[/tex] and [tex]\( (8x - 48) \)[/tex].

2. Factor out the greatest common factor from each group:
- In the first group [tex]\( (x^3 - 6x^2) \)[/tex], the greatest common factor is [tex]\( x^2 \)[/tex]. Factoring it out gives us:
[tex]\[ x^2(x - 6) \][/tex]
- In the second group [tex]\( (8x - 48) \)[/tex], the greatest common factor is [tex]\( 8 \)[/tex]. Factoring it out gives us:
[tex]\[ 8(x - 6) \][/tex]

3. Combine the groups: Now our expression looks like this:
[tex]\[ x^2(x - 6) + 8(x - 6) \][/tex]

4. Factor out the common binomial factor: Notice that [tex]\( (x - 6) \)[/tex] is a common factor in both terms. Factor [tex]\( (x - 6) \)[/tex] out:
[tex]\[ (x^2 + 8)(x - 6) \][/tex]

So, the factored form of the polynomial [tex]\( x^3 - 6x^2 + 8x - 48 \)[/tex] is [tex]\( (x^2 + 8)(x - 6) \)[/tex].

Therefore, the correct choice is:
- A. [tex]\( x^3 - 6x^2 + 8x - 48 = (x^2 + 8)(x - 6) \)[/tex]

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