Find the roots of the polynomial equation.
x3 – 2x2+ 10x + 136 = 0
the answer is 3 plus or minus sign 5i, –4
please show work
To find the roots of the polynomial equation, we can use various methods such as factoring, synthetic division, or using the quadratic formula. In this case, the quadratic formula will be the most efficient method.
The given polynomial equation is:
x^3 – 2x^2 + 10x + 136 = 0
To apply the quadratic formula, we first need to rewrite the equation in the form ax^2 + bx + c = 0. Let's divide the equation by the coefficient of x^3 to simplify it:
(x^3 – 2x^2 + 10x + 136)/1 = 0
x^3 – 2x^2 + 10x + 136 = 0
Now, we can identify a = 1, b = -2, and c = 10. The quadratic formula states that the roots of an equation in the form ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4(1)(10))) / (2(1))
x = (2 ± √(4 - 40)) / 2
x = (2 ± √(-36)) / 2
x = (2 ± 6i) / 2
Simplifying further:
x = 2/2 ± 6i/2
x = 1 ± 3i
Therefore, the complex roots are 1 + 3i and 1 - 3i.
To find the remaining real root, we can use synthetic division or by guessing/estimating possible real roots. By observing the cubic polynomial, we notice that x = -4 is a solution.
Using synthetic division:
-4 | 1 - 2 10 136
| -4 24 -136
---------------
1 - 6 34 0
The resulting polynomial is 1x^2 - 6x + 34. We can solve this equation separately using the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(1)(34))) / (2(1))
x = (6 ± √(36 - 136)) / 2
x = (6 ± √(-100)) / 2
x = (6 ± 10i) / 2
x = (3 ± 5i)
Therefore, the remaining root is 3 + 5i and 3 - 5i.
In conclusion, the roots of the polynomial equation x^3 – 2x^2+ 10x + 136 = 0 are 3 ± 5i and -4.