To factor the polynomial [tex]\( x^3 + 2x^2 + x \)[/tex] completely, follow these steps:
1. Identify a common factor:
Look for the greatest common factor in all the terms of the polynomial. The terms are [tex]\( x^3 \)[/tex], [tex]\( 2x^2 \)[/tex], and [tex]\( x \)[/tex]. The variable [tex]\( x \)[/tex] is present in all the terms, so [tex]\( x \)[/tex] is the greatest common factor.
2. Factor out the common factor:
After factoring [tex]\( x \)[/tex] out of each term, we get:
[tex]\[
x^3 + 2x^2 + x = x(x^2 + 2x + 1)
\][/tex]
3. Factor the quadratic:
Next, consider the quadratic [tex]\( x^2 + 2x + 1 \)[/tex]. We need to factor this quadratic expression. Notice that the quadratic can be factored using the perfect square trinomial pattern [tex]\( a^2 + 2ab + b^2 = (a + b)^2 \)[/tex].
In this case, [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex]:
[tex]\[
x^2 + 2x + 1 = (x + 1)^2
\][/tex]
4. Combine the factors:
Substitute back the factored form of the quadratic into our earlier expression:
[tex]\[
x(x^2 + 2x + 1) = x(x + 1)^2
\][/tex]
So, the completely factored form of [tex]\( x^3 + 2x^2 + x \)[/tex] is:
[tex]\[
x(x + 1)^2
\][/tex]
