MSC #18: The First by the First and the Last by the Last
“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) 2x^2 − 6x + 6; B) 2x^2 − 7x + 6; C) 2x^2 + 7x + 6; D) 2x^2 + 5x + 6; E) 2x^2 − 5x + 6
This is a question given in the senior and open levels of the 2nd International Vedic Mathematics Olympiad (IVMO 2022) which was held on November 26, 2022.
The easiest way to find other factors of the given 4th degree polynomial requires the use of theVedic Math sub-Sutra which is the subject of this post.
The Vedic Math sub-Sutra, First by the First and the Last by the Last, which we abbreviated as FFLL in our book, “Algebra Made Easy as Arithmetic” has varied applications in Mathematics. It can be used in full or in parts.
In estimating products or quotients, we often use only “the first by the first”- multiplying only the first digits, while we use “the last by the last” when in adding whole numbers in a column, we align the numbers by their last digits. Divisibility tests of 2, 4, 8, 5, 10 and some other numbers require that we look at only at the last digits.
Certain multiplication short cuts are developed if the first and last digits follow certain rules. Two-digit square roots and cube roots of perfect squares and cubes can be easily identified using FFLL.
In combination with another sub-Sutra, the Product of the Sums is the Sum of the Product, FFLL becomes a powerful tool for checking the results of arithmetic and algebraic calculations, factoring, and determining square roots of perfect square polynomials.
For example, if it is given that 5,041 is a perfect square, its square root can be easily determined by partitioning the square in groups of two digits starting from the right. The last digit of the square is 1, so the last digit of the square root must be either 1 or 9. The first two digits, 50, is much nearer to 49, the square of 7 than to 64, the square of 8. The square root must be 71.
Our illustration shows how we can quickly determine the other factor by just applying the First by the First and the Last by the Last.
A quick look at the given polynomials in our illustrated example will tell us that the other factor is a binomial. Dividing the first term of the 3rd degree polynomial by the first term of the trinomial, 6x3 ÷ 2x2 will give the first term of the binomial factor, 3x while dividing the last terms 12 ÷ 3 will give the last term of the binomial, 4. So (3x + 4) is the other factor.
Applying this FFLL technique in the IVMO question, we can see that the 1st term of the other factor is 6x^4 ÷ 3x^2 = 2x^2 and the last term would be -30 ÷ -5 = 6. Thus, the other factor would be in the form of (2x^2 + Bx + 6). Looking at the choices, we can see that all five of them is in the form mentioned earlier. So, another step is necessary. This step, the Product of the Sums is the Sum of the Products, which we “nicknamed” PSSP, is the topics of our 19th MATH-Inic Special for Christmas.