You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
Find the roots of the polynomial equation
- Thread starter Stutterr
- Start date
Find the roots of the polynomial equation
x^3-2x^2+10x+136=0 can someone explain how to do this and solve with imaginary numbers?
What have you tried? What do you know about finding roots? What class are you in? What methods are you familiar with?
This function has one real (and rational) root. Start by finding that. (If you graph it, the real root should be obvious.)
Knowing that root gives you a factor of the polynomial. Factor that out and you'll be left with a quadratic expression. Use the quadratic formula to find the other two (imaginary) roots.
Here is another approach if you like. We can factor if we can rearrange.
Rewrite it as:
\(\displaystyle x^{3}-6x^{2}+4x^{2}+34x-24x+136\)
\(\displaystyle x^{3}-6x^{2}+34x+4x^{2}-24x+136\)
Now, group:
\(\displaystyle (x^{3}-6x^{2}+34x)+(4x^{2}-24x+136)\)
Factor out a common term from each set of parentheses:
\(\displaystyle x(x^{2}-6x+34)+4(x^{2}-6x+34)\)
Notice what is in the parentheses is the same. This is essential when factoring:
\(\displaystyle (x+4)(x^{2}-6x+34)=0\)
Obviously, \(\displaystyle x=-4\) is a solution. Now, the quadratic formula can be used on the remaining quadratic:
\(\displaystyle x=\frac{-(-6)\pm\sqrt{(-6)^{2}-4(1)(34)}}{2(1)}=\frac{6\pm\sqrt{-100}}{2}\)
\(\displaystyle =\frac{6\pm 10i}{2}\)
\(\displaystyle x=3+5i, \;\ x=3-5i\)
Rewrite it as:
\(\displaystyle x^{3}-6x^{2}+4x^{2}+34x-24x+136\)
\(\displaystyle x^{3}-6x^{2}+34x+4x^{2}-24x+136\)
Now, group:
\(\displaystyle (x^{3}-6x^{2}+34x)+(4x^{2}-24x+136)\)
Factor out a common term from each set of parentheses:
\(\displaystyle x(x^{2}-6x+34)+4(x^{2}-6x+34)\)
Notice what is in the parentheses is the same. This is essential when factoring:
\(\displaystyle (x+4)(x^{2}-6x+34)=0\)
Obviously, \(\displaystyle x=-4\) is a solution. Now, the quadratic formula can be used on the remaining quadratic:
\(\displaystyle x=\frac{-(-6)\pm\sqrt{(-6)^{2}-4(1)(34)}}{2(1)}=\frac{6\pm\sqrt{-100}}{2}\)
\(\displaystyle =\frac{6\pm 10i}{2}\)
\(\displaystyle x=3+5i, \;\ x=3-5i\)