Now we have two terms x^2(x - 6) and -1(5x + 12) that can be further factored: x^2(x - 6) - 1(5x + 12) = x^2(x - 6) - 1(5x + 12) = x^2(x - 6) - 1(5x + 12)
Therefore, the solutions are x = 0, x = 6 and x = -12/5.
Rational Zero Theorem: We can also apply the Rational Zero Theorem to find the rational roots of the cubic equation.
The possible rational roots are all factors of the constant term divided by all factors of the leading coefficient: Factors of -12: ±1, ±2, ±3, ±4, ±6, ±12 Factors of -1: ±1
The possible rational roots are: ±1, ±2, ±3, ±4, ±6, ±12
We can test these possible roots using synthetic division or substitution to find the exact roots of the equation.
To solve the equation (-x^3) + (6x^2) - 5x - 12 = 0, we can use factoring or another method like the Rational Zero Theorem.
Factoring:We can first rearrange the equation:
-x^3 + 6x^2 - 5x - 12 = 0
Then, we can factor by grouping:
-x^3 + 6x^2 - 5x - 12 = (-x^3 + 6x^2) + (-5x - 12)
= x^2(-x + 6) - 1(5x + 12)
= x^2(x - 6) - 1(5x + 12)
Now we have two terms x^2(x - 6) and -1(5x + 12) that can be further factored:
x^2(x - 6) - 1(5x + 12)
= x^2(x - 6) - 1(5x + 12)
= x^2(x - 6) - 1(5x + 12)
Therefore, the solutions are x = 0, x = 6 and x = -12/5.
Rational Zero Theorem:We can also apply the Rational Zero Theorem to find the rational roots of the cubic equation.
The possible rational roots are all factors of the constant term divided by all factors of the leading coefficient:
Factors of -12: ±1, ±2, ±3, ±4, ±6, ±12
Factors of -1: ±1
The possible rational roots are: ±1, ±2, ±3, ±4, ±6, ±12
We can test these possible roots using synthetic division or substitution to find the exact roots of the equation.