Solution to Brainteaser No. 2

SOLUTION TO BRAINTEASER NO. 2 DISCLAIMER This solution is but one of the many possible methods for solving this problem. Various other solutions may lead to the same answers. This specific solution was provided by one of the members of the University of the Philippines Civil Engineering Society.

PROBLEM:

The polynomial P has four roots: 1/4, 1/2, 2, and 4. The product of the roots is 1 and P(1) = 1. Find P(0).

SOLUTION: Normally, determining the value of a function for a given value of x requires that the function be defined first; and that is exactly where this solution begins. Let us define the polynomial in its factored form P(x) = m (x – ¼) (x – ½) (x – 2) (x – 4) where m is an unknown constant, which we will solve for using the given conditions in the problem. We use the parameter P(1) = 1 to determine the value of m: 1

1

𝑃(1) = 𝑚 (1 − 4) (1 − 2) (1 − 2) (1 − 4) = 1 3

1

⇒ 𝑚 (4) (2) (−1) (−3) = 1 ⇒

9 𝑚 8

=1 8

⇒𝑚 = 9

As a quick solution, since the product of the roots is equal to 1, substituting 0 for all the x in the equation without considering m would effectively yield the product of the negatives of the roots, which is also coincidentally – having an even number of roots – equal to 1. This solution is presented in detail below: 𝑃(0) = = =

ANSWER:

8 1 1 (0 − 4) (0 − 2) (0 − 2) (0 − 9 8 1 1 (− 4) (− 2) (−2) (−4) 9 8 (1) 9

4)

P(0) = 8/9

UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños