Final answer:
To form a polynomial with a zero of -4 and a multiplicity of 3, one would use the factor (x + 4)^3. Expanding this factor results in the polynomial P(x) = x^3 + 12x^2 + 48x + 64, which is a third-degree polynomial with the given zero and multiplicity.
Step-by-step explanation:
To form a polynomial whose zeros and degree are given, we first need to understand the concept of zeros and multiplicity. The zero of a polynomial is a value of x for which the polynomial equals zero. If -4 is a zero with multiplicity 3, it means that (x + 4) is a factor of the polynomial three times.
The general form of a polynomial factor with a given zero and multiplicity is (x - c)^m, where c is the zero and m is the multiplicity. Since the zero given is -4 and its multiplicity is 3, the corresponding factor of the polynomial will be (x - (-4))^3, which simplifies to (x + 4)^3.
Therefore, the polynomial with -4 as a zero with multiplicity 3 is given by:
P(x) = (x + 4)^3
Expanding this using the binomial theorem or by multiplying (x + 4) by itself three times, we get:
P(x) = x^3 + 12x^2 + 48x + 64
This is the required polynomial of degree 3, since the highest exponent of x is 3. The degree of the polynomial is equal to the sum of the multiplicities of its zeros. In this case, we have only one zero, -4, with a multiplicity of 3, which gives us a third-degree polynomial.
The process outlined here is how a polynomial is formed based on its zeros and their multiplicities, and is a fundamental concept in algebra.