1.) What are the possible rational roots of the function f(x) = 3x^4 - x - 2
2.) What are the rational roots of the following equation: x^3 - 4x^2 - 4x +16 = 0
Thank you for any help!
1. 4 possible.
2. x = -2, 2, 4.
1. By the rule of Des Cartes, the coefficients changed signs once, so there is at least one real positive root, perhaps three.
The fact that sum of the coefficients equals zero, indicate that (x-1) is a factor.
Do a long division to reduce the polynomial to a cubic, and search for other rational roots, if there is any.
The dominance of the 3x^4 term tells us that the graph would resemble that of 3x^4, displaced -2 downwards, hence there would likely to be a positive root and a negative root, and two complex roots.
2. Factorize, if possible.
To find the possible rational roots of a polynomial equation, you can use the Rational Root Theorem. The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation with integer coefficients, then p must be a factor of the constant term (the last term) and q must be a factor of the leading coefficient (the coefficient of the highest-degree term).
1.) For the function f(x) = 3x^4 - x - 2, the constant term is -2 and the leading coefficient is 3. The factors of -2 are ±1 and ±2, and the factors of 3 are ±1 and ±3. Therefore, the possible rational roots are all the combinations of these factors, which are: ±1/1, ±1/3, ±2/1, ±2/3.
2.) For the equation x^3 - 4x^2 - 4x + 16 = 0, the constant term is 16 and the leading coefficient is 1. The factors of 16 are ±1, ±2, ±4, ±8, and ±16, and the factors of 1 are ±1. Therefore, the possible rational roots are all the combinations of these factors, which are: ±1/1, ±2/1, ±4/1, ±8/1, ±16/1.
To check if any of these roots are actually the solutions to the equations, you can use various methods like factoring, synthetic division, or numerical methods.