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If f (x) = x^3- 4x² – 25x + 28 and f(7) = 0, then find all of the zeros of f(x) algebraically.

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Final answer:

The zeros of the function f(x) are x = 7, x = -4, and x = 1.

Step-by-step explanation:

To find the zeros of the function f(x), we set f(x) equal to zero and solve for x. Given that f(7) = 0, we substitute 7 for x in the function and solve for the zeros.

f(x) = x^3 - 4x^2 - 25x + 28

f(7) = 7^3 - 4(7)^2 - 25(7) + 28

0 = 343 - 196 - 175 + 28

0 = 0

The equation holds true for x = 7, which means 7 is a zero of the function. To find the other zeros, we can divide the function by (x - 7) to get the remaining polynomial.

(x^3 - 4x^2 - 25x + 28) / (x - 7) = x^2 + 3x - 4

Now we can factor the remaining polynomial, x^2 + 3x - 4, to find the other zeros.

x^2 + 3x - 4 = (x + 4)(x - 1)

Therefore, the zeros of the function f(x) are x = 7, x = -4, and x = 1.

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