To completely factor the given polynomial expression [tex]\(x^5 + 7x^3 + 12x\)[/tex], let's go through a detailed, step-by-step solution.
Step 1: Identify and factor out the greatest common factor (GCF)
First, observe that each term in the polynomial [tex]\(x^5 + 7x^3 + 12x\)[/tex] contains the variable [tex]\(x\)[/tex]. Therefore, we can factor out an [tex]\(x\)[/tex] from each term.
[tex]\[ x^5 + 7x^3 + 12x = x(x^4 + 7x^2 + 12) \][/tex]
Now we have:
[tex]\[ x(x^4 + 7x^2 + 12) \][/tex]
Step 2: Factor the remaining polynomial
Next, we need to factor the polynomial inside the parentheses, [tex]\(x^4 + 7x^2 + 12\)[/tex]. Notice that this polynomial can be treated as a quadratic in terms of [tex]\(x^2\)[/tex].
Let [tex]\(y = x^2\)[/tex]. Then [tex]\(x^4 = y^2\)[/tex], and the polynomial becomes:
[tex]\[ y^2 + 7y + 12 \][/tex]
We need to factor this quadratic expression. To do this, look for two numbers that multiply to 12 (the constant term) and add to 7 (the coefficient of the middle term). The numbers 3 and 4 satisfy these conditions:
[tex]\[ y^2 + 7y + 12 = (y + 3)(y + 4) \][/tex]
Substituting [tex]\(y = x^2\)[/tex] back in, we get:
[tex]\[ (x^2 + 3)(x^2 + 4) \][/tex]
Step 3: Combine the factored terms
Don't forget the [tex]\(x\)[/tex] that we factored out initially. Therefore, the complete factorization of the original polynomial is:
[tex]\[ x(x^2 + 3)(x^2 + 4) \][/tex]
Final Answer
The completely factored form of the polynomial [tex]\(x^5 + 7x^3 + 12x\)[/tex] is:
[tex]\[ \boxed{x(x^2 + 3)(x^2 + 4)} \][/tex]
