# 30MSC 2023 #18: FFLL and PSSP

The First by the First(FFLL) and the Product of the Sums is the Sum of the Product(PSSP) are two Vedic Maths sub-sutras that, when used in combination, can be used to easily solve a variety of problems.

We will use this, problem, taken from the 2nd International Vedic Mathematics Olympiad (IVMO 2022) Senior questionaire, Given that (3x^2 + 3x – 5) is a factor of (6x^4 – 9x^3  − 7x^2  +  43x – 30) , which of the following is another factor? a) (2×2 − 6x +6); b) (2×2 − 7x +6); c) (2×2 +7x +6); d) (2×2 +5x +6);  e) (2×2 − 5x +6) to show how to use these two sub-sutras.

The usual solution to this type of problem is to actually perform polynomial division, but this is not easy considering that we will have a quadratic divisor. Using FFLL and PSSP will enable us to mentally get the correct answer.

Divide the first term of the fourth-degree polynomial by the first term of the given factor to find the first term of the other factor: 6x^4/3x^2 = 2x^2 (the first by the first). Then divide the last term of the longer polynomial by the last term of the given factor to get the last term of the other factor: -30/-5 = 6 ( the last by the last).

The other factor must then be (2x^2 + bx + 6). We can see that all the choices have the same first and last terms. We will now use the product of the sums is the sum of the product to find out the value of b inthe unknown factor.

Using only the coefficients, we have

(3 + 3 – 5) (2 + b + 6) = (6 – 9 – 7 + 43 – 30)

(8 + b) = (49 – 46)

(8 + b) = 3

b = – 5

The other factor is (2x^2 – 5x + 6).

The use of the FFLL and PSSP pair in other types of problems are discussed on Chapter 8 of our forthcoming book, 30 Master Strategies in Computing Vol II.

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #19: Using the Ekadhika