Write the general polynomial q(x) whose only zeroes are -3 and 7, with multiplicities 3 and 7 respectively. What is its degree?

Answer 1

We're given a polynomial with zeros -3 and 7.

This means that the equations x=-3 and x=7 are true.

If we move everything to one side, we get the equations x+3=0 and x-7=0. These are our factors for our polynomial equation, q(x) = (x+3)(x-7).

However, we're not done yet. Since we're given that the zeros -3 and 7 have multiplicities 3 and 7 respectively, that means that in our general polynomial equation, those respective factors are raised to those powers:

q(x) = (x+3)^3 * (x-7)^7

If we expand each of the factors, the first terms (which would have the highest degrees) would be x^3 and x^7.

Now, when we multiply those two expanded factors, the first two terms would make the term with the highest degree: x^3 * x^7 = x^10 (think FOIL, the first term is multiplying the two first terms).

Since we multiplied the two highest degree terms from the factors, it gives us our new highest degree term, x^10.

So, the general polynomial is

q(x) = (x+3)^3 * (x-7)^7

and the degree is 10.

Answer 2

Q(x)=(x+3)^3*(x-7)^7

Has degree 10