What is a cubic function that has real zeros at x = 3 and x = 7?

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To find a cubic function that has real zeros at x = 3 and x = 7, we need to formulate a function that includes these roots. A cubic function can generally be represented in the form:

f(x) = a(x – r1)(x – r2)(x – r3)

Here, a is a non-zero constant, and r1, r2, and r3 are the roots of the function. In our case, we know two of the roots:

  • r1 = 3
  • r2 = 7
  • r3 is yet to be determined (it can be any real number).

Let’s say we choose r3 = r, where r can be any real number. The function can then be written as:

f(x) = a(x – 3)(x – 7)(x – r)

For simplicity, let’s assume a = 1 (you can multiply it by any constant later). The function becomes:

f(x) = (x – 3)(x – 7)(x – r)

Now, if we specifically want a cubic polynomial with real coefficients, we can choose a real number for r. For instance, if we set r = 0, the equation simplifies to:

f(x) = (x – 3)(x – 7)(x)

Expanding this gives us:

f(x) = x(x - 3)(x - 7)
     = x[(x^2 - 10x + 21)]
     = x^3 - 10x^2 + 21x

Thus, a cubic function that has real zeros at x = 3 and x = 7, with a third zero at x = 0, is:

f(x) = x^3 – 10x^2 + 21x

This cubic function meets the requirement of having the specified real zeros.

In summary, there are an infinite number of cubic functions with zeros at x = 3 and x = 7. By selecting different values for r, you can generate various cubic functions while maintaining the same specified roots.

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