Answer :

We want to factor the expression

[tex]$$6x^3 - 48x^2 - x + 8.$$[/tex]

A good strategy is to group the terms into two pairs and factor out the common factors in each pair. Follow these steps:

1. Group the terms:

Write the expression as two groups:

[tex]$$ (6x^3 - 48x^2) + (-x + 8). $$[/tex]

2. Factor each group:

- In the first group, factor out the common factor [tex]$6x^2$[/tex]:

[tex]$$ 6x^3 - 48x^2 = 6x^2(x - 8). $$[/tex]

- In the second group, notice that [tex]$-x + 8$[/tex] can be factored by taking out [tex]$-1$[/tex]:

[tex]$$ -x + 8 = -1(x - 8). $$[/tex]

3. Extract the common binomial:

Now the expression becomes:

[tex]$$ 6x^2(x - 8) - 1(x - 8). $$[/tex]

Since both terms contain the factor [tex]$(x - 8)$[/tex], factor it out:

[tex]$$ (x - 8)(6x^2 - 1). $$[/tex]

4. Write the completely factored form:

The expression [tex]$6x^3 - 48x^2 - x + 8$[/tex] factors completely to:

[tex]$$ (x - 8)(6x^2 - 1). $$[/tex]

Thus, the fully factored form is:

[tex]$$\boxed{(x-8)(6x^2-1)}.$$[/tex]

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Rewritten by : Barada