what are the roots for x^3-13x-12?
x3-13x+12/x-1
To find the roots of the equation x^3 - 13x - 12, we can use a method called the "Rational Root Theorem" combined with synthetic division.
According to the Rational Root Theorem, any rational root of the equation must be of the form p/q, where p is a factor of the constant term (-12 in this case), and q is a factor of the leading coefficient (1 in this case).
The factors of -12 are ±1, ±2, ±3, ±4, ±6, ±12, and the factors of 1 are ±1.
Now we can use synthetic division with possible rational roots to find the actual roots:
1. Synthetic division using the root 1:
1 | 1 0 -13 -12
| 1 1 -12
---------------
1 1 -12 -24
As the remainder is not 0, 1 is not a root.
2. Synthetic division using the root -1:
-1 | 1 0 -13 -12
| -1 1 12
----------------
1 -1 -12 0
The remainder is 0, meaning -1 is a root of the equation.
Therefore, the roots of the equation x^3 - 13x - 12 are:
-1 and two other roots that can be found by solving the quadratic equation obtained from the synthetic division as follows:
1x^2 - 1x - 12 = 0
Factoring this quadratic equation, we get:
(x + 3)(x - 4) = 0
Thus, the other two roots are:
x = -3 and x = 4
Therefore, the roots of the equation x^3 - 13x - 12 are -1, -3, and 4.
To find the roots of a cubic equation like x^3 - 13x - 12 = 0, you can use various methods such as factoring, graphing, or using numerical methods like Newton's method. Here, I will explain how to find the roots using the Rational Root Theorem and synthetic division:
Step 1: Apply the Rational Root Theorem
The Rational Root Theorem states that if a rational number p/q is a root of the cubic equation, then p should be a factor of the constant term (-12 in our case), and q should be a factor of the leading coefficient (1 in our case).
The factors of -12 are ±1, ±2, ±3, ±4, ±6, and ±12.
The factors of 1 are ±1.
So, the potential rational roots are:
±1, ±2, ±3, ±4, ±6, ±12
Step 2: Apply synthetic division
Synthetic division is a way to simplify polynomial division. You can divide the polynomial by a potential root and see if the remainder is zero. If it is, then the number you divided by is a root.
Apply synthetic division for each potential root until you find a root that makes the remainder equal to zero.
For example, let's try dividing by x = 1:
1 | 1 0 -13 -12
--------------------
| 1 1 -12
The remainder is -12, not zero. Therefore, x = 1 is not a root.
Let's try x = -1:
-1 | 1 0 -13 -12
---------------------
-1 1 12
The remainder is also nonzero (-12). Hence, x = -1 is not a root.
You can continue trying the other potential rational roots using synthetic division until you find the root that gives a remainder of zero. Repeat the process until you find all the roots.
In this case, the rational roots are x = 3 and x = -4. To find the remaining root, you can use the quadratic formula on the remaining quadratic equation obtained after synthetic division.
I hope this explanation helps you understand the process of finding the roots of a cubic equation!
Y = x^3 - 13x - 12 = 0.
X = -1, x+1 = 0.
(x^3 - 13x - 12)/(x+1) = x*2 - x - 12.
x^3-13x-12 = (x+1)(x^2-x-12).
Factor the Quadratic:
(x+1)(x+3)(x-4) = 0.
x+3 = 0, X = -3.
x-4 = 0, X = 4.
The roots are: X = -1, -3, and 4.