I need a little help with this problem.
P(x)=2x^4-17x^3+47x^2-58x+24
(A) Find all possible candidates for real zeroes. I got +-1, +-2, +-3, +4, +-6,+-8, +-12, +-1.5, +-.5, and +-24.
Then I have to find all real zeroes. I tried all of the possible candidates and none of them worked. Could you please tell me where I may have gone wrong and if there is an easier way to find real zeros?
Thank you!
You mean rational zeroes. You applied the Rational Roots theorem to find all possible rational roots correctly, but you may have made a mistake when checking if P(x) is zero. But then, it may be that there are no rational zeroes. So, the fact that none of the candidades yields zero is not proof that a mistake has been made.
There are a few ways you can simplify the problem. The reason why you got so many possibilities is because 24 has so many divoisors. So, you could do a change of variables, e.g.:
x = t + 1
Then the constant term of the polynomial as a function of t will be the value at t = 0, which corresponds to x = 1, which is -2. So, that's a huge simplification.
If we define Q(t) = P(t+1)
Then to coefficient of t^4 will be 2, the constant term will be P(1) = -2
The Rational Roots theorem then yields the possible roots:
t = 2, -2, 1, -1, 1/2, -1/2
and therefore:
x = t+1 = 3, -1, 2, 0, 3/2, 1/2
We know that x = 0 is not possible, so the list has shrunk to:
x = 3, -1, 2, 3/2, 1/2
I checked them all (quickly, I could have made mistakes), and none of them worked, so it seems that there are no rational solutions.
Thank you so much for your help!
Finding the possible candidates for real zeroes is the first step to finding the real zeroes of a polynomial. However, your list of possible candidates appears to be incorrect. We can use the Rational Root Theorem to determine the possible candidates.
The Rational Root Theorem states that if a rational number P/Q is a root of a polynomial equation P(x) = 0, then P must be a factor of the constant term and Q must be a factor of the leading coefficient.
In this case, the constant term is 24 (the coefficient of x^0) and the leading coefficient is 2 (the coefficient of x^4). So, the possible rational roots are the factors of 24 divided by the factors of 2.
The factors of 24 are: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.
The factors of 2 are: ±1 and ±2.
So, the possible candidates for real zeroes are: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.
Now, to find the real zeroes, we can use a method called synthetic division or factor theorem. Let me explain:
1. Choose one of the possible candidates as a test root. Let's start with 1.
2. Use synthetic division to divide the polynomial P(x) by (x - 1).
The coefficients of P(x) are: 2, -17, 47, -58, 24, and we divide by (x - 1).
3. Perform synthetic division to obtain the remainder.
The result of the synthetic division will give you the quotient and the remainder. If the remainder is zero, then the test root is a real zero. If the remainder is nonzero, then the test root is not a real zero.
Continue this process with each of the possible rational roots until you find the real zeroes.
If none of the possible candidates work, it is possible that the real zeroes are irrational or complex. In such cases, you may need to use numerical methods, such as graphing the function or using algorithms like Newton's method.
I hope this clarifies the process for finding real zeroes and helps you identify the correct candidates for this particular polynomial. Let me know if you have any further questions!