find all the zeroes of:
x^5-3x^4-24x^3-72x^2-25x+75=0
5x^5-20x^4-40x^3-16x^2-45x+180=0
-3x^5+3x^4+9x^3-7x^2+12x=12
x^3-x^2-13x-13=0
x^3+x^2=15x-15
Thank you for any help...I'm desperate please help :(
For the first three, you would have to apply the rational zero theorem then use synthetic division to see if the remainder is zero.
For the last two, you would group the terms and factor out a common factor method(http://www.chesapeake.edu/khennayake/MAT113/3.2Solution.htm).
This was done by trial and error assisted by EXCEL. Very tedious!
1.Solution Set: X =-1.5887,0.7772,-1.2089, and 0.8717.
2.X = -2.07316,1.2139,-5.0432, and 2.2307.
3. X = -1.9548,2.1191, and 0.9050.
4. X = -2.218,4.516,-11.908,-1.0917.
5. x^3+x^2 = 15x-15.
x^3+x^2-15x+15 = 0.
NOTE: All values of X shown are called
ZEROES, because they give zero output
when plugged into the Eq.
I hope I've helped.
X = -2.218,1.220,13.923,1.0774.
To find the zeros (or roots) of a polynomial equation, we need to solve for the values of x that make the equation equal to zero. There are different methods to solve polynomial equations, such as factoring, synthetic division, or using numerical methods like Newton's method.
Let's go through each equation and explain how to find the zeros:
1. Equation: x^5 - 3x^4 - 24x^3 - 72x^2 - 25x + 75 = 0
Unfortunately, this equation is of degree 5, which means there is no general formula for finding its roots algebraically. The best approach is to use numerical methods or approximation techniques to find the roots. One possible approach is using the Newton-Raphson method, which requires an initial guess and iterative calculations to converge to the roots. However, using software or tools like Wolfram Alpha or math libraries in programming languages (e.g., Python's numpy or scipy) can automate this process and give you the approximate values of the roots.
2. Equation: 5x^5 - 20x^4 - 40x^3 - 16x^2 - 45x + 180 = 0
Similar to the previous equation, there is no straightforward algebraic way to find the roots of a quintic polynomial. The recommended approach is to use numerical methods or approximation techniques.
3. Equation: -3x^5 + 3x^4 + 9x^3 - 7x^2 + 12x = 12
We can start by simplifying the equation to: -3x^5 + 3x^4 + 9x^3 - 7x^2 + 12x - 12 = 0
There doesn't appear to be a simple factoring method, so we can resort to using numerical methods or approximation techniques. One possible approach is to use synthetic division to test for potential rational roots and then solve for the remaining roots using numerical methods.
4. Equation: x^3 - x^2 - 13x - 13 = 0
We can try factoring by grouping in this equation. Rearrange the equation as: (x^3 - x^2) - (13x + 13) = 0
Factor out common terms in each binomial: x^2(x - 1) - 13(x + 1) = 0
Rearrange further: x^2(x - 1) + (-13)(x + 1) = 0
Factor out (x - 1): (x - 1)(x^2 - 13) + (-13)(x + 1) = 0
At this point, one of the factors is x - 1. We can apply the quadratic formula to solve for x^2 - 13 = 0, which gives us x = ±√13.
Therefore, the zeros of the equation are x = 1, x = √13, and x = -√13.
5. Equation: x^3 + x^2 = 15x - 15
Rearrange the equation: x^3 + x^2 - 15x + 15 = 0
Again, there isn't an apparent factoring method for this equation. We can use numerical methods or approximation techniques to find the roots.
In conclusion, for complex polynomial equations, numerical methods or approximation techniques are usually employed to find the zeros. However, for equations with lower degrees, techniques like factoring, synthetic division, and the quadratic formula can be helpful.