The factored form of the polynomial is:
(x−4)(x−2)(x+3)
Let's factor the polynomial:
x^3 −3x^2−10x+24
We can factor the polynomial by first finding one factor using the rational root theorem, and then factoring the remaining quadratic.
Steps to solve:
1. Find one factor using the rational root theorem:
The rational root theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must have a numerator that is a factor of the constant term, and a denominator that is a factor of the leading coefficient.
In this case, the constant term is 24, and the leading coefficient is 1. Therefore, the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.
We can try plugging these values into the polynomial to see if they are roots. We find that x = 4 is a root of the polynomial.
2. Factor the remaining quadratic:
We can factor the remaining quadratic using the quadratic formula:
x= −b± √b^2−4ac/ 2a
In this case, a = 1, b = 1, and c = -6. Plugging these values into the formula, we get:
x= −1± √1^2 −4(1)(−6)/2(1)
x= −1± √25/2
x= −1±5 / 2
x=2 or x=−3
Therefore, the factored form of the polynomial is:
(x−4)(x−2)(x+3)