Multiplying a Monomial by a Monomial
Multiplying two monomials
We begin with 4 × x = x + x + x + x =
Similarly, 4 × (3x) = 3x + 3x + 3x + 3x =
Now, observe the following products.
(i) x × 3y = x × 3 × y = 3 × x × y =
(ii) 5x × 3y = 5 × x × 3 × y = 5 × 3 × x × y =
(iii) 5x × (–3y) = 5 × x × (–3) × y = 5 × (–3) × x × y =
Notice that all the three products of monomials: 3xy, 15xy, –15xy are also monomials.
Some more useful examples follow.
(i) 5x × 4 x 2
Separate out the variables and constants and multiply them individually
So, 5x × 4 x 2 x × x 2
Thus the answer is 20 x 3
(ii) 5x × (– 4xyz)
Separate out the variables and constants and multiply them individually
So, 5x × − 4 xyz = (5×4) × (− x × xyz ) =
Thus the answer is − 20 xyz .
(ii)Try these
Note that 5 × 4 =
i.e., coefficient of product = coefficient of first monomial × coefficient of second monomial;
and x × x 2
Thus, algebraic factor of product = algebraic factor of first monomial × algebraic factor of second monomial.
Multiplying three or more monomials
Observe the following examples.
(i) 4xy ×
- The algebra expression is 4xy ×
5 x 2 y 2 6 x 3 y 3 - multiply with x terms and y terms
x 3 y 3 x 3 y 3 - Divide the terms and multiply with values
( x 3 x 3 y 3 y 3 - Calculate the x terms and y terms separately 120
x
It is clear that We, first multiply the first two monomials and then multiply the resulting monomial by the third monomial. This method can be extended to the product of any number of monomials.
Example 3
Complete the table for area of a rectangle with given length and breadth.
Solution
| length | breadth | area |
|---|---|---|
| 3x | 5y | 3x × 5y = |
| 9y | 9y x 4 | |
| 4ab | 5bc | 4ab x 5bc = |
Example 4
Find the volume of each rectangular box with given length, breadth and height.
| No | length | breadth | height |
|---|---|---|---|
| (i) | 2ax | 3by | 5cz |
| (ii) | |||
| (iii) | 2q | 4 | 8 |
Hence, for (i) volume = (2ax) × (3by) × (5cz)
Volume = 2 × 3 × 5 × (ax) × (by) × (cz) =
for (ii) volume = m 2 n 2 p 2 m 2 n 2 p 2
for (iii) volume = 2q x 4q 2 q 3 q 2 q 3 q 6