To find the completely factored form of the polynomial [tex]\(3x^5 - 7x^4 + 6x^2 - 14x\)[/tex], let's walk through a step-by-step factorization process:
1. Identify and extract the greatest common factor (GCF):
First, note that each term in the polynomial has at least one [tex]\( x \)[/tex]. We can factor out an [tex]\( x \)[/tex] from each term:
[tex]\[
3x^5 - 7x^4 + 6x^2 - 14x = x(3x^4 - 7x^3 + 6x - 14)
\][/tex]
2. Factor the resulting polynomial further:
Now we need to check if we can factor the polynomial [tex]\( 3x^4 - 7x^3 + 6x - 14 \)[/tex] further. Since it is a quartic polynomial (degree 4), finding its roots or special factorization is more complex.
3. Look for any special factorizations such as patterns or possibilities to group terms:
Sometimes, polynomials can be factored by grouping, but in this case, we do not have an obvious factorization by grouping or special patterns.
4. Consider irreducible polynomial factors:
If it’s not easily factorizable by common methods and advanced techniques (such as using the Rational Root Theorem, synthetic division, or others), we conclude that the polynomial form we initially produced may be in its simplest form in terms of common factorization available through elementary algebra methods.
Given the structure and complexity of the given polynomial and verifying against standard polynomial factorization techniques, the factored form of the polynomial [tex]\(3x^5 - 7x^4 + 6x^2 - 14x\)[/tex] remains as:
[tex]\[
x(3x^4 - 7x^3 + 6x - 14)
\][/tex]
So, the completely factored form of [tex]\(3x^5 - 7x^4 + 6x^2 - 14x\)[/tex] is:
[tex]\[
x(3x^4 - 7x^3 + 6x - 14)
\][/tex]
