I'm working with finding roots of polynomial equations with degrees of 3 or higher. I have the equation
r(x)=x^4-6x^3+12x^2=6x-13
I used a graphing calculator to find the real roots of 1,-1
Then I did synthetic using -1, and I ended up with the equation
x^3-6x^2+2x-13
How do I get this to a quadratic so that I can find the imaginary roots? A friend suggested grouping, but I'm not real good at grouping, and I couldn't figure out how to factor it or group it. Any help would be appreciated.
Thanks,
Josh
*in the original equation it was supposed to be +6x-13 not =6x-13
3 < b + a X 7
To find the quadratic equation from the given cubic equation, you can use the method called synthetic division. Here's how you can do it:
1. Start with the cubic equation: x^3 - 6x^2 + 2x - 13.
2. Since you already found that -1 is a root of the original equation, perform synthetic division using -1 as the divisor. Write down the coefficients of the cubic equation: 1, -6, 2, -13.
-1 | 1 -6 2 -13
|___ ____ ___ ____
-1 7 -9 22
3. The result of synthetic division gives us a new quadratic equation: x^2 + 7x - 9, with a remainder of 22.
Therefore, the quadratic equation equivalent to the given cubic equation x^3 - 6x^2 + 2x - 13 is x^2 + 7x - 9. Now, you can use various methods to find the roots of this quadratic equation, including factoring, completing the square, or using the quadratic formula.