Answer:
Apologies for any confusion. Here's an example of a polynomial of degree 5 with the given zeros x = -1, 7, 6, and 3:
P(x) = (x + 1)(x - 7)(x - 6)(x - 3)(x - 3)
This polynomial has the given zeros and a degree of 5.
Explanation:
To find a polynomial of degree 5 with the given zeros x = -1, 7, 6, and 3, we can use the zero-product property and the fact that if a is a zero of a polynomial, then (x - a) is a factor of that polynomial.
Given zeros: x = -1, 7, 6, 3
To find a polynomial of degree 5, we need five linear factors. Using the given zeros, the polynomial can be expressed as:
P(x) = (x - (-1))(x - 7)(x - 6)(x - 3)(x - a)
To determine the last factor (x - a), we need one more zero. Since the degree of the polynomial is 5, we have one remaining zero.
Let's arbitrarily choose a = 2. Then the polynomial becomes:
P(x) = (x - (-1))(x - 7)(x - 6)(x - 3)(x - 2)
Expanding this polynomial will yield a degree-5 polynomial with the given zeros:
P(x) = (x + 1)(x - 7)(x - 6)(x - 3)(x - 2)
Note that there are multiple correct answers since we can choose different values for the last zero (x - a) as long as it is not one of the given zeros.