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 (an-k) / (an) where a is the coefficient (an is the first coefficient, an-1 is the second, etc.), k is number of roots being added, and n is the degree of the polynomial.
