Sure, let's solve the polynomial [tex]\( 3x^3 - 24x^2 - 60x = 0 \)[/tex] by factoring step-by-step.
1. Factor out the greatest common factor (GCF):
The polynomial [tex]\( 3x^3 - 24x^2 - 60x \)[/tex] has a common factor of [tex]\( 3x \)[/tex]. So, we can factor out [tex]\( 3x \)[/tex] from each term:
[tex]\[
3x^3 - 24x^2 - 60x = 3x(x^2 - 8x - 20)
\][/tex]
2. Factor the quadratic expression:
Now we need to factor the quadratic expression [tex]\( x^2 - 8x - 20 \)[/tex].
To do this, we look for two numbers that multiply to [tex]\(-20\)[/tex] (the constant term) and add up to [tex]\(-8\)[/tex] (the coefficient of [tex]\( x \)[/tex]).
Those numbers are [tex]\( -10 \)[/tex] and [tex]\( 2 \)[/tex] because:
[tex]\[
(-10) \times 2 = -20 \quad \text{and} \quad (-10) + 2 = -8
\][/tex]
Therefore, we can factor the quadratic as:
[tex]\[
x^2 - 8x - 20 = (x - 10)(x + 2)
\][/tex]
3. Combine the factored terms:
Now substituting back our factorization into the equation, we get:
[tex]\[
3x(x - 10)(x + 2) = 0
\][/tex]
4. Solve for the roots:
According to the Zero Product Property, if the product of multiple factors equals zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
3x = 0 \quad \Rightarrow \quad x = 0
\][/tex]
[tex]\[
x - 10 = 0 \quad \Rightarrow \quad x = 10
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
Therefore, the roots of the polynomial [tex]\( 3x^3 - 24x^2 - 60x = 0 \)[/tex] are:
[tex]\[
x = 0, x = 10, \text{and } x = -2
\][/tex]
So, the correct answer is:
[tex]\[
c. -2,0,10
\][/tex]
