Finding degree of a polynomial written in factored form

Question

Find the degree , zeroes and multiplicities of the function f(x) = x^4 (7x^2 - 63)^5 (x+4)^6 (1-x)^3 (x^3 + 8)

 

I got (-16807x^26) as the leading term so the degree is 26.

 

For zeroes I got:

0 mutl. of 4

3  mult. of 5

-3 mult. of 5

-4 mult. of 6

1 mult. of 3

-2 mult. of 3

 

I don't know if I'm doing it correctly. The quadratic binomial factor raised to the 5th power is throwing me off. I can't find similar example in my textbook.

 

Thank you

Answer 1

You're right about the degree. Even without multiplying it out, the degree must be 4 + 2*5 + 6 + 3 + 3 = 26.

 

You're also correct about the zeros. Notice that we can factor:

 

(7x^2 - 63)⁵ = (7(x^2-9))⁵ = (7(x-3)(x+3))⁵ = 7⁵(x-3)⁵(x+3)⁵