IB exam question: Let p(x)=2x^5+x^4–26x^3–13x^2+72x+36, x∈R. For the polynomial equation p (x) = 0 , state (i) the sum of the roots; (ii) the product of the roots.

p(x) = 2x⁵ + x⁴ – 26x³ – 13x² + 72x + 36
i) obtain the sum by using vieta's fomula: for p(x) the sum is hence -b/a hence, sum = -1/2
ii) obtain the product by using the fomula -f/a. This is because the p(x) is an odd degree polynomial, hence, there is a negative. If p(x) was an even degree the product would be just f/a. hence, product = -36/2 = -18 Note: The IB will change the degree of the polynomials and may ask for diffrent combination of addition and multiplication of the roots e.g. for some third degree polynomial f(x) find x₁x₂ + x₁x₃ + x₂x₃. It is important, therfore, to memorise the general formula for adding roots: (-1)^k (a_n-k) / (a_n) where a is the coefficient (a_n is the first coefficient, a_n-1 is the second, etc.), k is number of roots being added, and n is the degree of the polynomial.