Final answer:
To find a polynomial of degree 5 with given zeros, we can use the factored form of a polynomial. The factored form is f(x) = a(x - r1)(x - r2)(x - r3)(x - r4)(x - r5), where r1, r2, r3, r4, and r5 are the zeros.
Step-by-step explanation:
To find a polynomial of degree 5 with the given zeros, we can use the factored form of a polynomial. The factored form is given by f(x) = a(x - r1)(x - r2)(x - r3)(x - r4)(x - r5), where r1, r2, r3, r4, and r5 are the zeros and a is a constant. In this case, the zeros are -7, -7, -1, 6, and 8. So the polynomial becomes f(x) = a(x + 7)(x + 7)(x + 1)(x - 6)(x - 8).
Now, we just need to find the value of a. Since the degree of the polynomial is 5, we know that the coefficient of the highest power term is non-zero. So, we can set up an equation using one of the known zeros and solve for a. Let's use -7 as the zero:
0 = a(-7 + 7)(-7 + 7)(-7 + 1)(-7 - 6)(-7 - 8)
0 = a(0)(0)(-6)(-13)(-15)
0 = a(0)
Therefore, a can be any value except 0. So the polynomial f(x) = a(x + 7)(x + 7)(x + 1)(x - 6)(x - 8), where a is a non-zero constant.