find all zeros of x^4-5x^3+10^2-20x+24 (leave all irrational answers in radical form)
i simply don't understand this question, i can find the rational value but never can i calculate how to find the irrational values nor the imaginary number values. what mistake am I making ?
I am pretty sure you forgot the x in 10^2 and it should have been x^4-5x^3+10x^2-20x+24
let f(x) = x^4-5x^3+10x^2-20x+24
try x = ±1, ±2, ±3, etc, that is, factors of 24
I found f(2) and f(3) = 0
so (x-2) and (x-3) would be factors.
Doing synthetic division consecutively by x-2 and x-3 gave me
x^4-5x^3+10^2-20x+24
= (x-2)(x-3)(x^2 + 4)
set x^2 + 4 = 0
x^2 = -4
x = ±2i
so zeros are 2, 3, 2i, and -2i
Thank you sooooo much !!!
To find all the zeros of a polynomial, including both rational and irrational or imaginary values, you can use a combination of polynomial long division, factoring, and the quadratic formula. Let's go through the steps to find the zeros of the given equation:
Step 1: Factor the polynomial as much as possible. In this case, the polynomial does not appear to have a straightforward factorization. So we need to proceed with other methods.
Step 2: Determine if there are any rational roots using the rational root theorem. The rational root theorem states that any rational root of a polynomial equation with integer coefficients must be of the form p/q, where p is a factor of the constant term (in this case, 24), and q is a factor of the leading coefficient (in this case, 1). Start by listing the factors of 24: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. To check if any of these are zeros, substitute them into the equation and see if the equation equals zero. Proceed with synthetic division to check each potential root until one works. If a rational root is found, it means that the polynomial can be factored using this root. However, if none of the potential roots work, it does not necessarily mean there are no rational roots. It only means that you need to explore other methods further.
Step 3: If you were successful in finding a rational root, you can now use polynomial long division or synthetic division to factor the polynomial further. Dividing the polynomial by (x - r), where r is the rational root found, will yield a reduced polynomial with one degree less. Repeat this process with the reduced polynomial until you obtain a quadratic.
Step 4: Solve the resulting quadratic equation using the quadratic formula. For a quadratic equation in the form ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). Substitute the values of a, b, and c from the quadratic equation and simplify the expression to find the solutions. These solutions will give you irrational or imaginary values if they are not rational.
Note: If at any point during these steps, you come across complex numbers (involving the imaginary unit i), it means that there are no real solutions, and the zeros lie in the complex number plane.
Remember to carefully check your calculations and simplifications at each step to avoid errors.
If you have a specific polynomial you'd like me to help you find the zeros for, please provide it, and I'll guide you through the steps.