Answer:
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Explanation:
The zeros, also known as roots or x-intercepts, of a polynomial function are the values of x for which the function is equal to zero. To find the zeros of the function f(x) = 7x^2 - x^3, we can set the function equal to zero and solve for x:
f(x) = 7x^2 - x^3 = 0
7x^2 = x^3
x^3 - 7x^2 = 0
This is a polynomial equation, so we can use the factor theorem or synthetic division to factor the polynomial and find the zeros.
Using the factor theorem, we can write the polynomial as (x - r)(x^2 + ax + b) = 0, where r is a zero of the polynomial and a and b are constants. Substituting the given polynomial and setting the expression equal to zero, we get:
(x - r)(x^2 + ax + b) = x^3 - 7x^2 = 0
Matching coefficients, we can find that a = 0 and b = 7. Thus, the polynomial can be written as:
(x - r)(x^2 + 0x + 7) = 0
(x - r)(x^2 + 7) = 0
The zeros of the polynomial are the roots of the equation (x - r)(x^2 + 7) = 0. The roots of this equation are x = r and x = sqrt(7) and x = -sqrt(7).
The graph of the function f(x) = 7x^2 - x^3 will cross the x-axis at x = r and x = sqrt(7) and x = -sqrt(7).