Multiplication of Algebraic Expression


In multiplication of algebraic expression before taking up the product of algebraic expressions, let us look at two simple rules.

(i) The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative.

(ii) if x is a variable and m, n are positive integers, then

(xᵐ × xⁿ) = x\(^{m + n}\)


Thus, (x³ × x⁵) = x⁸, (x⁶ + x⁴) = x\(^{6 + 4}\) = x\(^{10}\), etc. 


I. Multiplication of Two Monomials

Rule:

Product of two monomials = (product of their numerical coefficients) × (product of their variable parts)

Find the product of: (i) 6xy and -3x²y³

Solution:

(6xy) × (-3x²y³)

= {6 × (-3)} × {xy × x²y³}

= -18x\(^{1 + 2}\) y\(^{1 + 3}\)

= -18x³y⁴.

(ii) 7ab², -4a²b and -5abc

Solution:

(7ab²) × (-4a²b) × (-5abc)

= {7 × (-4) × (-5)} × {ab² × a²b × abc}

= 140 a\(^{1 + 2 + 1}\) b\(^{2 + 1 + 1}\) c

= 140a⁴b⁴c.

II. Multiplication of a Polynomial by a Monomial

Rule:

Multiply each term of the polynomial by the monomial, using the distributive law a × (b + c) = a × b + a × c.

Find each of the following products:

(i) 5a²b² × (3a² - 4ab + 6b²)

Solution:

5a²b² × (3a² - 4ab + 6b²)

= (5a²b²) × (3a²) + (5a²b²) × (-4ab) + (5a²b²) × (6b²)

= 15a⁴b² - 20a³b³ + 30a²b⁴.

(ii) (-3x²y) × (4x²y - 3xy² + 4x - 5y)

Solution:

(-3x²y) × (4x²y - 3xy² + 4x - 5y)

= (-3x²y) × (4x²y) + (-3x²y) × (-3xy²) + (-3x²y) × (4x) + (-3x²y) × (-5y)

= -12x⁴y² + 9x³y³ - 12x³y + 15x²y².

III. Multiplication of Two Binomials

Suppose (a + b) and (c + d) are two binomials. By using the distributive law of multiplication over addition twice, we may find their product as given below.

(a + b) × (c + d)

= a × (c + d) + b × (c + d)

= (a × c + a × d) + (b × c + b × d)

= ac + ad + bc + bd

Note: This method is known as the horizontal method. 


(i) Multiply (3x + 5y) and (5x - 7y).

Solution:

(3x + 5y) × (5x - 7y)

= 3x × (5x - 7y) + 5y × (5x - 7y)

= (3x × 5x - 3x × 7y) + (5y × 5x - 5y × 7y)

= (15x² - 21xy) + (25xy - 35y²)

= 15x² - 21xy + 25xy - 35y²

= 15x² + 4xy - 35y².

Column wise multiplication

The multiplication can be performed column wise as shown below.

    3x + 5y

× (5x - 7y)
_____________
  15x² + 25xy                  ⇐ multiplication by 5x.

           - 21xy - 35y²       ⇐ multiplication by -7y.
__________________
15x²  +  4xy  - 35y²       ⇐ multiplication by (5x - 7y).
__________________

(ii) Multiply (3x² + y²) by (2x² + 3y²)

Solution:

Horizontal method,

= 3x² (2x² + 3y²) + y² (2x² + 3y²)

= (6x⁴ + 9x²y²) + (2x²y² + 3y⁴)

= 6x⁴ + 9x²y² + 2x²y² + 3y⁴

= 6x⁴ + 11x²y² + 3y⁴

Column methods,

     3x² +  y²

× (2x² +  3y³)
_____________
    6x⁴ +  2x²y²                  ⇐ multiplication by 2x² .

          +  9x²y² + 3y⁴        ⇐ multiplication by 3y³.
___________________
  6x⁴ + 11x²y² + 3y⁴        ⇐ multiplication by (2x² + 3y³).
___________________

IV. Multiplication by Polynomial

We may extend the above result for two polynomials, as shown below.

(i) Multiply (5x² – 6x + 9) by (2x -3)

    5x² – 6x + 9

×         (2x - 3)
____________________
   10x³ - 12x² + 18x               ⇐ multiplication by 2x.

          - 15x² + 18x - 27         ⇐ multiplication by -3.
______________________
 10x³ – 27x² + 36x - 27         ⇐ multiplication by (2x - 3).
______________________

Therefore, (5x² – 6x + 9) by (2x - 3) is 10x³ – 27x² + 36x – 27

(ii) Multiply (2x² – 5x + 4) by (x² + 7x – 8)

Solution:

By column method

    2x² –  5x + 4

×  (x² +  7x – 8)
___________________________
   2x⁴ –   5x³ +   4x²                       ⇐ multiplication by x².

         + 14x³ - 35x² + 28x              ⇐ multiplication by 7x.

                   - 16x² + 40x - 32         ⇐ multiplication by -8.
___________________________
 2x⁴ –   9x³ - 47x² + 68x - 32         ⇐ multiplication by (x² + 7x - 8).
___________________________

Therefore, (2x² – 5x + 4) by (x² + 7x – 8) is 2x⁴ – 9x³ - 47x² + 68x – 32.

(iii) Multiply (2x³ – 5x² – x + 7) by (3 - 2x + 4x²)

Solution:

Arranging the terms of the given polynomials in descending power of x and then multiplying,

    2x³ – 5x² – x + 7

×      (3 - 2x + 4x²)
_________________________________
   8x⁵ - 20x⁴ –   4x³ + 28x²                      ⇐ multiplication by 3.

         -  4x⁴ + 10x³ +   2x² – 14x             ⇐ multiplication by -2x.

                  +   6x³ – 15x² -   3x + 21      ⇐ multiplication by 4x².
_________________________________
 8x⁵ – 24x⁴ + 12x³ + 15x² – 17x + 21    ⇐ multiplication by (3 - 2x + 4x²).
_________________________________


 Algebraic Expression

Algebraic Expression

Addition of Algebraic Expressions

Subtraction of Algebraic Expressions

Multiplication of Algebraic Expression

Division of Algebraic Expressions









8th Grade Math Practice 

From Multiplication of Algebraic Expression to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  2. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  3. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  4. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More

  5. Worksheet on Multiplying 1-Digit Numbers |Multiplying One Digit Number

    Mar 25, 24 03:39 PM

    Multiplication tables will help us to solve the worksheet on multiplying 1-digit numbers. The questions are based on multiplying one digit number and word problems on multiplying one digit number.

    Read More