# 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 2^{nd} 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**