Special Multiplication Methods for Polynomials

a. Criss-Cross Multiplication

• Example 1: Consider the multiplication of the binomials: (3�+2)(3x+2) and (4�−5)(4x−5).

• Using the FOIL method:
  • First: 3y x 4y= 12y²
  • Outer: 3y x -5= −15x
  • Inner: 2 x 4y = 8y
  • Last: 2 x −5= −10

  Combining the results: (3y + 2)(4y − 5) = 12y² – 15y + 8y – 10

• Example 2: Let’s multiply (2y − 3) and (y + 4)
• Using the FOIL method:
  • First: 2y x y = 2y²
  • Outer: 2y x 4 = 8y
  • Inner: -3 x y = -3y
  • Last: -3 x 4 = -12

  Combining the results: (2y – 3)(y+4) = 2y² + 8y – 3y – 12 = 2y² + 5y – 12

b. Vertical and Crosswise

• Step 1: Write the coefficients of the two polynomials vertically, with each term in its own column.

• Step 2: Draw diagonal lines from each term of the first polynomial to each term of the second polynomial, forming a lattice or grid.

• Step 3: Multiply the coefficients at the intersections of the diagonals, moving from right to left and top to bottom.

• Step 4: Add up the partial products along the diagonals to get the final result, combining like terms if necessary.

Let’s illustrate this method with an example:

Example:

• Consider multiplying the polynomials (3x + 2) and (2x – 1) using the vertical and crosswise method.
• Write the coefficients of the polynomials in a grid:

3 2

——-

2 | |

-1 | |

• Draw diagonals and calculate partial products:

3 2

——-

2 | 6 4 |

-1 | -3 -2 |

——-

• Add up the partial products along the diagonals:

• The diagonal starting from the bottom-right corner has partial products 4 and -2, summing up to 2.

• The diagonal starting from the top-right corner has partial products 6 and -3, summing up to 3.

• So, the final result is 3x^2 + 2x + 3.

So, the product of (3x + 2) and (2x – 1) is 3x^2 + 2x + 3 using the vertical and crosswise method for polynomials. This method offers a systematic approach to polynomial multiplication, making it easier to manage and compute, especially for complex expressions.

Vedic Mathematics is an ancient Indian system of mathematical techniques. It offers efficient and ingenious methods for solving mathematical problems. One of the fascinating aspects is its approach to multiplication, especially when dealing with polynomials. In this article, we will explore the special multiplication methods in Vedic Maths that make multiplying polynomials a breeze.