How can I find a cubic function with the given zeros of 6, 3, and 5?

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To find a cubic function given the zeros (or roots) of the function, you can use the fact that a polynomial can be constructed from its roots.

The given zeros are 6, 3, and 5. This means that the cubic function can be represented in factored form as:

f(x) = (x - r1)(x - r2)(x - r3)

Where r1, r2, and r3 are the roots of the function. In this case, we substitute:

f(x) = (x - 6)(x - 3)(x - 5)

Next, we need to expand this expression to obtain the standard form of the cubic polynomial.

  1. First, multiply the first two factors:
    2. (x - 6)(x - 3) = x2 - 9x + 18
  2. Now, take this result and multiply it by the third factor:
    3. f(x) = (x2 - 9x + 18)(x - 5)

Now we distribute the terms:

= x3 - 5x2 - 9x2 + 45x + 18x - 90
= x3 - 14x2 + 63x - 90

So, the cubic function with the given zeros of 6, 3, and 5 is:

f(x) = x3 - 14x2 + 63x - 90

And that’s the cubic function you were looking for! You can verify that the roots are indeed 6, 3, and 5 by substituting these values into the function and confirming that they yield zero.

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