Answer :
To factor the polynomial [tex]\( x^3 + 8x^2 + 6x + 48 \)[/tex] by grouping, we will look at the terms in pairs and factor out the greatest common factor from each pair.
1. Group the terms:
[tex]\[
(x^3 + 8x^2) + (6x + 48)
\][/tex]
2. Factor out the greatest common factor from each group:
- For the first group [tex]\( x^3 + 8x^2 \)[/tex], the greatest common factor is [tex]\( x^2 \)[/tex]. Factoring out [tex]\( x^2 \)[/tex], we get:
[tex]\[
x^2(x + 8)
\][/tex]
- For the second group [tex]\( 6x + 48 \)[/tex], the greatest common factor is [tex]\( 6 \)[/tex]. Factoring out [tex]\( 6 \)[/tex], we get:
[tex]\[
6(x + 8)
\][/tex]
3. Write the expression with the common factor:
After factoring each group, the polynomial looks like this:
[tex]\[
x^2(x + 8) + 6(x + 8)
\][/tex]
4. Factor out the common binomial factor:
Notice that both terms contain the binomial [tex]\((x + 8)\)[/tex]. We can factor [tex]\((x + 8)\)[/tex] out:
[tex]\[
(x + 8)(x^2 + 6)
\][/tex]
Therefore, the factored form of the polynomial [tex]\( x^3 + 8x^2 + 6x + 48 \)[/tex] is [tex]\((x + 8)(x^2 + 6)\)[/tex].
1. Group the terms:
[tex]\[
(x^3 + 8x^2) + (6x + 48)
\][/tex]
2. Factor out the greatest common factor from each group:
- For the first group [tex]\( x^3 + 8x^2 \)[/tex], the greatest common factor is [tex]\( x^2 \)[/tex]. Factoring out [tex]\( x^2 \)[/tex], we get:
[tex]\[
x^2(x + 8)
\][/tex]
- For the second group [tex]\( 6x + 48 \)[/tex], the greatest common factor is [tex]\( 6 \)[/tex]. Factoring out [tex]\( 6 \)[/tex], we get:
[tex]\[
6(x + 8)
\][/tex]
3. Write the expression with the common factor:
After factoring each group, the polynomial looks like this:
[tex]\[
x^2(x + 8) + 6(x + 8)
\][/tex]
4. Factor out the common binomial factor:
Notice that both terms contain the binomial [tex]\((x + 8)\)[/tex]. We can factor [tex]\((x + 8)\)[/tex] out:
[tex]\[
(x + 8)(x^2 + 6)
\][/tex]
Therefore, the factored form of the polynomial [tex]\( x^3 + 8x^2 + 6x + 48 \)[/tex] is [tex]\((x + 8)(x^2 + 6)\)[/tex].
