Answer:
(a) To find the values of b and c, we can use the remainder theorem. According to the remainder theorem, if f(x) is divided by (x - a), the remainder will be equal to f(a).
Given that when f(x) is divided by (x - 1), the remainder is 0, we can substitute x = 1 into f(x) and set it equal to 0:
f(1) = 1^3 + 18^2 + b(1) + c = 0
Similarly, when f(x) is divided by (x + 1), the remainder is -8. Substituting x = -1 into f(x):
f(-1) = (-1)^3 + 18^2 + b(-1) + c = -8
Simplifying these equations, we have:
1 + 18^2 + b + c = 0 ...(1)
-1 + 18^2 - b + c = -8 ...(2)
To solve these equations, we can subtract equation (2) from equation (1):
2 + 2b = 8
Solving for b, we have:
2b = 6
b = 3
Substituting the value of b into equation (1), we can find the value of c:
1 + 18^2 + 3 + c = 0
c = -(1 + 18^2 + 3)
Hence, the values of b and c are 3 and -(1 + 18^2 + 3) respectively.
The full equation is f(x) = x^3 + 18^2 + 3x - (1 + 18^2 + 3).
(b) To factorize f(x) completely, we need to find the roots (zeros) of f(x). The roots are the values of x for which f(x) equals zero.
Let's set f(x) equal to zero and solve for x:
x^3 + 18^2 + 3x - (1 + 18^2 + 3) = 0
Simplifying, we have:
x^3 + 3x - 1 = 0
Unfortunately, this equation cannot be factored easily using simple techniques. We can use numerical methods or a graphing calculator to find the approximate roots.
(c) The roots (zeros) of f(x) are the values of x where f(x) equals zero. From the previous equation:
x^3 + 3x - 1 = 0
The roots can be found by solving this equation. However, as mentioned earlier, this equation cannot be easily factored or solved analytically.
To find the approximate roots, we can use numerical methods such as the Newton-Raphson method or use a graphing calculator. These methods will provide us with the approximate values of x where f(x) equals zero.
Explanation:
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