Question
Factor the polynomial:
3x^4 + 6x^3 + 9x^2
3x^4 + 6x^3 + 9x^2
Answers
GPT-4o mini
To factor the polynomial \(3x^4 + 6x^3 + 9x^2\), we can start by identifying the greatest common factor (GCF) of the terms.
The GCF of the coefficients (3, 6, and 9) is 3. Additionally, each term contains at least \(x^2\). Therefore, we can factor out \(3x^2\):
\[
3x^4 + 6x^3 + 9x^2 = 3x^2(x^2 + 2x + 3)
\]
Now, we need to check if the quadratic \(x^2 + 2x + 3\) can be factored further. We can attempt to factor it by checking its discriminant:
The discriminant \(D\) of the quadratic \(ax^2 + bx + c\) is given by \(D = b^2 - 4ac\). Here, \(a = 1\), \(b = 2\), and \(c = 3\):
\[
D = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]
Since the discriminant is negative (\(-8\)), \(x^2 + 2x + 3\) has no real roots and cannot be factored over the reals.
Thus, the fully factored form of the given polynomial is:
\[
\boxed{3x^2(x^2 + 2x + 3)}
\]
The GCF of the coefficients (3, 6, and 9) is 3. Additionally, each term contains at least \(x^2\). Therefore, we can factor out \(3x^2\):
\[
3x^4 + 6x^3 + 9x^2 = 3x^2(x^2 + 2x + 3)
\]
Now, we need to check if the quadratic \(x^2 + 2x + 3\) can be factored further. We can attempt to factor it by checking its discriminant:
The discriminant \(D\) of the quadratic \(ax^2 + bx + c\) is given by \(D = b^2 - 4ac\). Here, \(a = 1\), \(b = 2\), and \(c = 3\):
\[
D = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]
Since the discriminant is negative (\(-8\)), \(x^2 + 2x + 3\) has no real roots and cannot be factored over the reals.
Thus, the fully factored form of the given polynomial is:
\[
\boxed{3x^2(x^2 + 2x + 3)}
\]