What is x to the fourth power minus 20x plus 64 equals 0
The equation you've provided is a polynomial equation. To solve it, we need to find the values of x that make the equation true, in other words, the values that satisfy the equation.
To begin, let's restate the equation:
x^4 - 20x + 64 = 0
To solve this equation, we can use a method called factoring, which involves factoring the expression on the left side of the equation and setting each factor equal to zero.
First, let's try to factor the expression:
x^4 - 20x + 64
To factor this expression, we can look for possible factors by considering the constant term (64) and the coefficient of the x term (-20).
In this case, the factorization might not be immediately obvious. So, we can look for rational roots using a technique called the Rational Root Theorem. According to this theorem, potential rational roots are of the form p/q, where p is a factor of the constant term (in this case, 64) and q is a factor of the leading coefficient of the highest degree term (in this case, 1, since the highest degree term is x^4).
The factors of 64 are:
±1, ±2, ±4, ±8, ±16, ±32, ±64
The factors of 1 are:
±1
So, the potential rational roots are:
±1, ±2, ±4, ±8, ±16, ±32, ±64
We can now check each potential root by substituting it into the equation.
By considering all the potential rational roots and evaluating the equation, we find that x = 2 is a root of the equation.
Once we've found one root, we can divide the equation by (x - 2) using polynomial long division or synthetic division to find the other roots.
Dividing the polynomial (x^4 - 20x + 64) by (x - 2), we get:
x^3 + 2x^2 - 16x - 32
Now, we can factor the new expression:
(x - 2)(x^3 + 2x^2 - 16x - 32) = 0
We can continue factoring the quadratic expression (x^3 + 2x^2 - 16x - 32) or use other methods to find the remaining roots.
By solving this quadratic expression, we find two more roots: x = -4 and x = 4.
So, the solutions to the equation x^4 - 20x + 64 = 0 are x = -4, x = 2, and x = 4.