Question

Explain the following

(i) How to multiply and divide rational expressions.

(ii) How to add and subtract rational expressions.

(iii) What LCD do we use to perform the addition in the expression $\frac{3}{x-1}+\frac{5}{x+2}$?

Short answer

(i) The way to multiply and divide rational expression is stated above.

(ii) The way to addition and subtraction of rational expression is stated above.

(iii) LCD of the given expression is$\left(x-1\right)\left(x-2\right)$

Step-by-step solution

Part i. Step 1. Write the given information.

The procedure to multiply and divide rational expression is required.

Part i. Step 2. Procedure to multiply and divide.

The multiplication of two rational expression is done by multiplying their numerators and multiplying their denominators.

Part i. Step 3. Procedure to multiply and divide.

To divide a rational expression by another expression, completely factor both numerator and denominator of all fraction, change the division sign to multiplication sign and reciprocate the fractions and reduce the fraction by cancelling the common fractions from the denominator and numerators.

Thus, the way to multiply and divide rational expression is stated above.

Part ii. Step 1. Write the given information.

The procedure to subtract and add rational expression is required.

Part ii. Step 2. Procedure for subtraction.

To solve the subtraction of the ration number, first of all the denominator should be checked whether it is having the common denominator or not.

If the denominator present in the rational number are same, then subtract the terms in the numerator directly.

If the denominators are different, then cross multiply the numerator and denominator, then write it as new numerator.

If common terms are there in numerator and denominator then cancel them and solve the rest subtraction.

Part ii. Step 3. Procedure for addition.

To solve the addition of the rational number, first of all the denominator should be checked whether it is having the common denominator or not.

If the denominator present in the rational number are same, then add the terms in the numerator directly.

If the denominators are different, then cross multiply the numerator and denominator, then write it as new numerator.

If common terms are there in numerator and denominator then cancel them and solve the rest follow the addition.

Thus, the way to addition and subtraction of rational expression is stated above.

Part iii. Step 1. Consider the given expression.

$\frac{3}{x-1}+\frac{5}{x+2}$

Part iii. Step 2. Procedure for LCD.

The multiplication of denominator of the rational number gives the lowest common divisor (LCD).

Part iii. Step 3. Find the LCD.

In the given expression, the LCD is,

$LCD=\left(x-1\right)\left(x+2\right)$

Thus, LCD of the given expression is$\left(x-1\right)\left(x-2\right)$