Answer :
Sure, let's factor the polynomial [tex]\( x^3 - 8x^2 + 6x - 48 \)[/tex] by grouping. Here’s a step-by-step solution:
1. Group the terms: We start by grouping the polynomial into two pairs:
[tex]\[
(x^3 - 8x^2) + (6x - 48)
\][/tex]
2. Factor out the greatest common factor from each group:
- First group: [tex]\( x^3 - 8x^2 \)[/tex]
The common factor here is [tex]\( x^2 \)[/tex].
[tex]\[
x^3 - 8x^2 = x^2(x - 8)
\][/tex]
- Second group: [tex]\( 6x - 48 \)[/tex]
The common factor here is [tex]\( 6 \)[/tex].
[tex]\[
6x - 48 = 6(x - 8)
\][/tex]
3. Rewrite the expression using these factored groups:
[tex]\[
x^2(x - 8) + 6(x - 8)
\][/tex]
4. Factor out the common binomial factor [tex]\((x - 8)\)[/tex]:
Both terms, [tex]\( x^2(x - 8) \)[/tex] and [tex]\( 6(x - 8) \)[/tex], have a common factor of [tex]\((x - 8)\)[/tex]. We factor [tex]\((x - 8)\)[/tex] out:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
Therefore, the polynomial [tex]\( x^3 - 8x^2 + 6x - 48 \)[/tex] can be factored by grouping as:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
This is the factored form of the polynomial.
1. Group the terms: We start by grouping the polynomial into two pairs:
[tex]\[
(x^3 - 8x^2) + (6x - 48)
\][/tex]
2. Factor out the greatest common factor from each group:
- First group: [tex]\( x^3 - 8x^2 \)[/tex]
The common factor here is [tex]\( x^2 \)[/tex].
[tex]\[
x^3 - 8x^2 = x^2(x - 8)
\][/tex]
- Second group: [tex]\( 6x - 48 \)[/tex]
The common factor here is [tex]\( 6 \)[/tex].
[tex]\[
6x - 48 = 6(x - 8)
\][/tex]
3. Rewrite the expression using these factored groups:
[tex]\[
x^2(x - 8) + 6(x - 8)
\][/tex]
4. Factor out the common binomial factor [tex]\((x - 8)\)[/tex]:
Both terms, [tex]\( x^2(x - 8) \)[/tex] and [tex]\( 6(x - 8) \)[/tex], have a common factor of [tex]\((x - 8)\)[/tex]. We factor [tex]\((x - 8)\)[/tex] out:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
Therefore, the polynomial [tex]\( x^3 - 8x^2 + 6x - 48 \)[/tex] can be factored by grouping as:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
This is the factored form of the polynomial.
