identify the number of zeros ....thank you
2x^4-250x=0
ALL polynomials of degree n have exactly n zeroes.
The Fundamental Theorem of Algebra
However, not all may be real. In this case,
2x^4 - 250x
= 2x(x^3-125) = 2x(x^3 - 5^3)
= 2x(x-5)(x^2+5x+25)
So, there are two real roots and two complex roots
To identify the number of zeros of the equation 2x^4 - 250x = 0, we first need to find the roots or values of x that satisfy the equation. Zeros, roots, and solutions are all the same thing; they represent the values of x that make the equation equal to zero.
To solve this equation, we can factor out the common factor of x:
x(2x^3 - 250) = 0
Now we have two factors: x and (2x^3 - 250). The equation will be equal to zero if either of these factors equals zero.
1. Setting x = 0:
If we substitute x = 0 into the equation, the first factor becomes zero:
0(2x^3 - 250) = 0
The entire equation becomes zero, so x = 0 is a solution or a zero of the equation.
2. Setting 2x^3 - 250 = 0:
To find the zeros of the second factor, we need to solve the equation 2x^3 - 250 = 0.
To solve this equation, we can isolate x by dividing both sides by 2:
(2x^3 - 250) / 2 = 0 / 2
x^3 - 125 = 0
Now, we can factorize the equation using the difference of cubes formula, which states that a^3 - b^3 can be factored as (a - b)(a^2 + ab + b^2). In this case, a = x and b = 5:
(x - 5)(x^2 + 5x + 25) = 0
Setting each factor equal to zero, we have two possibilities:
a) x - 5 = 0
Adding 5 to both sides:
x = 5
b) x^2 + 5x + 25 = 0
This quadratic equation does not have real solutions because its discriminant, b^2 - 4ac, is negative. Therefore, no real values of x satisfy this equation.
In summary, the equation 2x^4 - 250x = 0 has two zeros: x = 0 and x = 5.