Answer:
Therefore, the roots of the given equation are x = 2, x = 1, and x = -3.
Explanation:
This problem involves finding the zeros (or roots) of a polynomial equation, which are the values of x that make the equation equal to zero. The given equation is cubic, meaning it has a degree of 3 and can have up to three real roots.
One way to find the roots of this equation is to use the Rational Root Theorem, which states that any rational root of a polynomial equation with integer coefficients must have the form p/q, where p is a factor of the constant term (in this case 18) and q is a factor of the leading coefficient (in this case 3). However, this method only works for finding rational roots, and there may be irrational or complex roots as well.
Another method is to use a graphing calculator or software to graph the equation and visually locate the x-intercepts, which are the points where the graph crosses the x-axis and the value of y is zero. From the graph, we can see that there are three real roots: one positive, one negative, and one between -2 and -1.
A third method is to use numerical methods (such as Newton's method or the Bisection method) to estimate the roots to a desired level of accuracy. However, this method involves iterative calculations and can be time-consuming.
Without using a graphing calculator, we can try to factor the given equation by using the Rational Root Theorem. The possible rational roots are ±1, ±2, ±3, ±6, ±9, ±18 (all factors of 18 divided by all factors of 3). We can test these roots by substituting them into the equation and seeing if the result equals zero.
Testing x = 1 gives:
3(1)^3 - 2(1)^2 - 13(1) + 18 = 3 - 2 - 13 + 18 = 6, which is not zero.
Testing x = -1 gives:
3(-1)^3 - 2(-1)^2 - 13(-1) + 18 = -3 - 2 + 13 + 18 = 26, which is not zero.
Testing x = 2 gives:
3(2)^3 - 2(2)^2 - 13(2) + 18 = 3(8) - 2(4) - 13(2) + 18 = 0, which means x = 2 is a root.
Using polynomial division, we can factor out (x - 2) from the cubic polynomial to obtain a quadratic polynomial that can be factored:
(3x^3 - 2x^2 - 13x + 18) / (x - 2) = 3x^2 + 4x - 9
Factoring the quadratic gives:
3x^2 + 4x - 9 = (3x - 3)(x + 3)
Setting each factor equal to zero and solving for x gives:
3x - 3 = 0, so x = 1
x + 3 = 0, so x = -3