Question
Factor the polynomial: -5x^3 - 10x^2 - 15x
Answers
GPT-4o mini
To factor the polynomial \(-5x^3 - 10x^2 - 15x\), we can start by identifying the greatest common factor (GCF) of the terms.
The coefficients are -5, -10, and -15, and the GCF of these numbers is -5.
Now, let's also consider the variable \(x\): the lowest power of \(x\) that appears in all terms is \(x\) (i.e., \(x^1\)).
Thus, we can factor out \(-5x\) from the polynomial:
\[
-5x^3 - 10x^2 - 15x = -5x(x^2 + 2x + 3)
\]
Now we check if the quadratic \(x^2 + 2x + 3\) can be factored further. To see if it can be factored, we can look for two numbers that multiply to \(3\) (the constant term) and add to \(2\) (the coefficient of \(x\)). However, there are no such integer pairs.
Next, we can calculate the discriminant to analyze its factorability:
\[
D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]
Since the discriminant is negative, \(x^2 + 2x + 3\) cannot be factored into real numbers.
Thus, the complete factorization of the polynomial is:
\[
-5x(x^2 + 2x + 3)
\]
This is the final factored form.
The coefficients are -5, -10, and -15, and the GCF of these numbers is -5.
Now, let's also consider the variable \(x\): the lowest power of \(x\) that appears in all terms is \(x\) (i.e., \(x^1\)).
Thus, we can factor out \(-5x\) from the polynomial:
\[
-5x^3 - 10x^2 - 15x = -5x(x^2 + 2x + 3)
\]
Now we check if the quadratic \(x^2 + 2x + 3\) can be factored further. To see if it can be factored, we can look for two numbers that multiply to \(3\) (the constant term) and add to \(2\) (the coefficient of \(x\)). However, there are no such integer pairs.
Next, we can calculate the discriminant to analyze its factorability:
\[
D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]
Since the discriminant is negative, \(x^2 + 2x + 3\) cannot be factored into real numbers.
Thus, the complete factorization of the polynomial is:
\[
-5x(x^2 + 2x + 3)
\]
This is the final factored form.