Which correctly describes the roots of the following cubic equation?
x^3-5x^2+3x+9=0
(x+1)(x-3)(x-3) = 0
Three real roots
single real at x = -1
double real at x = 3
http://www.wolframalpha.com/input/?i=solve+x%5E3+-5x%5E2+%2B+3x+%2B+9+%3D+0+%3D+0
thank you
You are welcome.
Well, if I were to describe the roots of this cubic equation, I would say they are like a group of shy potatoes hiding underground, afraid to come out and face the world. In other words, the roots of this equation are real and possibly irrational numbers, but without further calculations, we can't determine their exact values. So, let's just imagine these roots as bashful spuds for now! 🥔
To find the roots of the cubic equation x^3 - 5x^2 + 3x + 9 = 0, we can use different methods such as factoring, synthetic division, graphing, or using numerical methods. Let's use the synthetic division method.
First, we need to guess a possible root of the equation. In this case, let's start with x = 1. We will need to check if it is indeed a root of the equation.
Substitute x = 1 into the equation:
(1)^3 - 5(1)^2 + 3(1) + 9 = 1 - 5 + 3 + 9 = 8
Since the result is not zero, x = 1 is not a root of the equation.
Now, let's try another value. We can use a process called the Rational Root Theorem to find possible rational roots. According to the theorem, the possible rational roots are the divisors of the constant term (9) divided by the divisors of the leading coefficient (1).
The divisors of 9 are ±1, ±3, ±9, and the divisors of 1 are ±1. Combining the possible divisors, we get ±1 and ±9.
We will try these possible rational roots by substituting them one by one into the equation:
1. Substitute x = -1:
(-1)^3 - 5(-1)^2 + 3(-1) + 9 = -1 - 5 - 3 + 9 = 0
Since the result is zero, x = -1 is a root of the equation.
Using synthetic division, divide the equation by (x + 1):
Coefficient of x^3: 1
Coefficient of x^2: -6
Coefficient of x: 9
Constant term: -9
The result of the synthetic division is:
x^2 - 6x + 9
Now, we have a quadratic equation x^2 - 6x + 9 = 0. This can be factored as (x - 3)(x - 3) = 0, where x = 3 is a repeated root. So, the roots of the cubic equation x^3 - 5x^2 + 3x + 9 = 0 are x = -1 (a single root) and x = 3 (a repeated root).