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) = x10, 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}

= -18x1 + 2 y1 + 3

= -18x3y4.

(ii) 7ab2, -4a2b and -5abc

Solution:

(7ab2) × (-4a2b) × (-5abc)

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

= 140 a1 + 2 + 1 b2 + 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

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

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