Multiplying Polynomials : Basic Techniques, Examples & Practice Questions

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.

Overview of Vedic Maths

Vedic Mathematics is a system of mathematical techniques that originated in ancient India. It’s based on ancient Hindu scriptures known as the Vedas and contains methods to solve complex mathematical problems quickly and efficiently.

The principles of Vedic Mathematics emphasize mental calculation, simplification of arithmetic operations, and finding patterns in numbers. It offers alternative approaches to traditional methods, making mathematical calculations more intuitive and easier to grasp.

Importance of Vedic Maths in Polynomial Multiplication

Vedic Mathematics offers several techniques that can be particularly useful for polynomial multiplication. Here’s why it’s important:

• Efficiency: Vedic Maths provides methods like criss-cross multiplication and vertical and crosswise multiplication, which can significantly reduce the number of steps required to multiply polynomials compared to traditional methods. This efficiency is especially beneficial when dealing with large or complex polynomials.

• Simplicity: The techniques in Vedic Mathematics are often simpler and more intuitive than conventional methods. This simplicity can make polynomial multiplication easier to understand and perform, even for those who may struggle with traditional algebraic approaches.

• Mental Calculation: Many Vedic Maths techniques are designed for mental calculation, allowing practitioners to perform polynomial multiplication mentally and without the need for pen and paper. This mental agility can be advantageous in various situations, such as exams or real-life problem-solving scenarios.

Basic Vedic Multiplication Techniques

Vertical and Crosswise Method (Nikhilam Sutra):

This method involves breaking down the numbers into parts and multiplying vertically and crosswise.

For example, let’s multiply 23 and 47:

23

x 47

_______

161 (23 * 7)

+ 940 (23 * 40)

_______

1081 (Final result)

Urdhva-Tiryagbhyam (Vertically and Crosswise):

In this technique, we multiply the digits vertically and crosswise.

For instance, let’s multiply 34 and 56:

3 4

x 5 x 6

________

15 12 (Cross multiplication)

20 24 (Vertical multiplication)

1904 (Final result)

Anurupyena (Proportionately):

This method involves adjusting one number relative to another to simplify multiplication.

For example, let’s multiply 37 by 11:

37

x 11

_______

407 (37 * 11)

Sunyam Samya Samuccaye (By the Completion of the Zeroes):

Identify zeroes in the multiplicands and multiplicands, and adjust them accordingly before multiplication.

For instance, let’s multiply 102 by 5:

102

x 5

_______

510 (102 * 5)

Ekadhikena Purvena (By One More than the Previous One):

In this technique, one of the multiplicands is incremented while the other is decreased by one.

For example, let’s multiply 13 by 14:

13

x 14

_______

12 (13 – 1)

14 (14 + 1)

—————–
182 (Final result)

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.

Solved Examples

Example 1: Criss-Cross Multiplication

• Polynomial A : 3x² + 2x + 5
• Polynomial B : 2x − 1
• Follow the criss-cross multiplication method to find the product.

Example 2: Vertical and Crosswise Multiplication

• Polynomial C : 4x² + 3x − 1
• Polynomial D : x + 2
• Use the vertical and crosswise method to compute the product.

Practice Questions

Below are a set of practice questions for readers to reinforce their understanding of the Vedic Maths methods for polynomial multiplication.

1. Perform multiplication using the criss-cross method:
   • 23×4123×41
   • 47×3247×32
   • 56×1956×19
2. Use vertical and crosswise multiplication to find the products:
   • 37×2437×24
   • 51×6851×68
   • 92×1392×13
3. Apply the lattice method to multiply the following numbers:
   • 78×2378×23
   • 65×4165×41
   • 83×7283×72
4. Utilize Vedic math techniques to multiply:
   • 47×1847×18
   • 59×2759×27
   • 86×1386×13
5. Practice criss-cross multiplication with:
   • 34×5634×56
   • 72×3972×39
   • 85×4785×47
6. Multiply 2x² + 3x + 1 by x − 4 using criss-cross multiplication.
7. Apply the vertical and crosswise method to find the product of 5x² + 2x − 3 and 3x + 1.

Interesting Facts about Vedic Maths

• Vedic Maths dates back to ancient Indian scriptures, with roots in the Vedas.
• The word “Veda” means knowledge, and Vedic Maths aims to provide systematic and easy-to-apply mathematical knowledge.
• Vedic Mathematics encompasses 16 Sutras (formulas) and 13 sub-Sutras, each designed to simplify specific mathematical operations.

Vedic Maths for Special Multiplication methods – FAQs

1. Can Vedic Maths be applied to more complex polynomial expressions?

Yes, Vedic Maths techniques can be adapted to handle more complex polynomials. The key is to understand the basic methods and build upon them for larger expressions.

2. Are Vedic Maths techniques only applicable to multiplication?

While Vedic Maths has a strong emphasis on multiplication, its principles can be extended to various mathematical operations, including division, addition, and subtraction.

3. How long does it take to master Vedic Maths for polynomial multiplication?

The time required to master Vedic Maths techniques varies from person to person. Regular practice and a solid understanding of the underlying principles are crucial for proficiency.

4. Can Vedic Maths be integrated into modern educational curricula?

Yes, many educators around the world are incorporating Vedic Maths into their teaching methods to enhance students’ mathematical skills and problem-solving abilities.

5. Are Vedic Maths methods widely recognized in academic circles?

While Vedic Maths may not be universally recognized in all academic circles, its popularity is growing, and many individuals find its methods valuable for mental calculation and problem-solving.