# Which quadratic function has real zeros at x = 3 and x = 7 and is cubic?

Polynomial functions are one of the most integral concepts of mathematics. They find their application in various fields of engineering and science. Depending on the degree of the polynomials, it can have one or more zeroes; while some may not have any root at all. Let's solve a problem related to these polynomials.

## Answer: The cubic function which has real zeros at x = 3 and x = 7 are x^{3} - 17x^{2} + 91x - 147 or x^{3} - 13x^{2} + 51x - 63.

Let's understand how we arrived at the solution.

**Explanation:**

Cubic polynomials can have 3 zeroes.

But we are given only two.

Hence, we need to have one of the roots with a multiplicity equal to 2.

Hence, this question can have two solutions:

Case 1: Cubic equation having roots x = 3, 3, 7.

Here we use x = 3 with a multiplicity equal to 2.

Therefore, the function is f(x) = (x - 3)(x - 3)(x - 7) = x^{3} - 13x^{2} + 51x - 63.

Case 1: Cubic equation having roots x = 3, 7, 7.

Here we use x = 7 with a multiplicity equal to 2.

Therefore, the function is f(x) = (x - 3)(x - 7)(x - 7) = x^{3} - 17x^{2} + 91x - 147.

Check out more on the multiplication of polynomials or you can verify this using multiplying polynomial calculator.