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]
[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]
