Final answer:
To find a polynomial with given zeros and degree, you can use factoring techniques, synthetic division application, and the rational root theorem. These techniques allow you to determine the factored form, degree, and additional zeros of the polynomial.
Step-by-step explanation:
To find a polynomial with given zeros and degree, you can use several techniques:
Factoring techniques: If you know the zeros of a polynomial, you can factor it using the zero-product property. For example, if the zeros are 2, -3, and 5, the factored form of the polynomial would be (x-2)(x+3)(x-5).
Synthetic division application: Synthetic division can be used to divide a polynomial by a linear factor corresponding to one of its zeros. By repeating this process, you can find the complete factored form of the polynomial.
Degree of polynomials: The degree of a polynomial is the highest power of the variable in the polynomial. By using the given zeros, you can determine the degree of the polynomial.
Rational root theorem usage: The rational root theorem states that if a polynomial has a rational root, it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By using this theorem, you can determine if the given zeros are rational and find additional zeros.