Question

Form a polynomial whose real zeros and degree are given.

Answer 1

To form a polynomial with given real zeros and degree, we can use the factored form of a polynomial. The polynomial can be written as P(x) = a(x - x1)(x - x2)...(x - xn), where a is a constant and n is the degree of the polynomial.

Step-by-step explanation:

To form a polynomial with given real zeros, we can use the factored form of a polynomial. Suppose we are given the real zeros x1, x2, ..., xn and the degree of the polynomial. The polynomial can be written as P(x) = a(x - x1)(x - x2)...(x - xn), where a is a constant and n is the degree of the polynomial.

For example, let's say we are given the real zeros 2 and -3, and the degree of the polynomial is 3. The polynomial can be formed as P(x) = a(x - 2)(x + 3)(x - r), where r is another real zero. The constant a and the real zero r can be determined from additional information related to the polynomial.

It's important to note that there are infinitely many polynomials that can have the same real zeros and degree, but with different coefficients and constant terms.

Answer 2

Given the roots of the polynomial below,

`-3,0,8`

with degree 3 and the leading coefficient of 1.

Therefore,

`f(x)=x(x-(-3))(x-8)`

Let us now expand the above function

`\begin{gathered} f(x)=x(x+3)(x-8) \\ f(x)=x(x(x-8)+3(x-8)) \\ f(x)=x(x^2-8x+3x-24) \\ f(x)=x(x^2-5x-24) \\ f(x)=x^3-5x^2-24x \end{gathered}`

Hence, the polynomial is

`f(x)=x^3-5x^2-24`