To factor the polynomial [tex]\( 25x^4 + 10x^3 + 5x^2 + 2x \)[/tex] completely, we need to follow a systematic approach.
1. Identify and factor out the greatest common factor (GCF):
The polynomial [tex]\( 25x^4 + 10x^3 + 5x^2 + 2x \)[/tex] has the GCF of [tex]\( x \)[/tex], since each term contains at least one [tex]\( x \)[/tex]. Factor [tex]\( x \)[/tex] out:
[tex]\[
25x^4 + 10x^3 + 5x^2 + 2x = x (25x^3 + 10x^2 + 5x + 2)
\][/tex]
2. Analyze the cubic polynomial inside the parentheses:
Next, we want to factor the cubic polynomial [tex]\( 25x^3 + 10x^2 + 5x + 2 \)[/tex]. We start by trying to group the terms in a way that can facilitate further factoring.
Observe that this polynomial does not have a common factor among all its terms, but we can attempt to factor by grouping or use known factorization techniques.
3. Attempt further factorization of the cubic polynomial:
We notice that [tex]\( 25 \)[/tex] can be expressed as [tex]\( 5^2 \)[/tex]. Let’s look for possible binomial factors:
We can test out some potential pairs:
[tex]\(
25x^3 + 10x^2 + 5x + 2 = (ax + b)(cx^2 + dx + e).
\)[/tex]
After careful consideration and comparing coefficients, we find that it factors nicely into:
[tex]\[
25x^3 + 10x^2 + 5x + 2 = (5x + 2)(5x^2 + 1)
\][/tex]
4. Complete the factorization:
Combining our factored terms, we have:
[tex]\[
25x^4 + 10x^3 + 5x^2 + 2x = x(5x + 2)(5x^2 + 1)
\][/tex]
Hence, the completely factored form of [tex]\( 25x^4 + 10x^3 + 5x^2 + 2x \)[/tex] is:
[tex]\[
x(5x + 2)(5x^2 + 1)
\][/tex]
