To factor the expression [tex]\( x^3 + 8x \)[/tex] completely, let's go through the steps methodically:
1. Identify the Common Factor:
First, observe if there is a common factor in the terms [tex]\( x^3 \)[/tex] and [tex]\( 8x \)[/tex].
Both terms have [tex]\( x \)[/tex] as a common factor.
So, we can factor out [tex]\( x \)[/tex]:
[tex]\[
x^3 + 8x = x(x^2 + 8)
\][/tex]
2. Examine the Remaining Polynomial:
After factoring out [tex]\( x \)[/tex], we are left with [tex]\( x^2 + 8 \)[/tex].
We need to determine if [tex]\( x^2 + 8 \)[/tex] can be factored further.
3. Check for Further Factorization:
[tex]\( x^2 + 8 \)[/tex] is a sum of squares, and typically, sums of squares cannot be factored over the real numbers.
4. Write the Complete Factorization:
Since [tex]\( x^2 + 8 \)[/tex] cannot be factored further using real numbers, our complete factorization is simply:
[tex]\[
x^3 + 8x = x(x^2 + 8)
\][/tex]
So, the completely factored form of the expression [tex]\( x^3 + 8x \)[/tex] is:
[tex]\[
\boxed{x(x^2 + 8)}
\][/tex]
