SUBTRACTING POLYNOMIALS

The following steps will be useful to subtract a polynomial from another polynomial. 

Step 1 :

Take the first polynomial as it is. Multiply all the terms of the second polynomial by negative sign. 

For example, if the polynomial B is subtracted from the polynomial A, then multiply all the terms of the polynomial B by negative sign.

Step 2 : 

After having multiplied all the terms of the second polynomial by negative sign, combine the like terms. 

Like Terms or Similar Terms

Like terms are the terms which have the same variables with same exponent for each variable.

Example :

7x, 3x, - 4x

Unlike or Dissimilar Terms

Unlike terms are the terms which have same variables or different variables. 

If they have same variables, the exponents will not be same. 

Example :

9x², 5xy, - 4xy², y, 6

More clearly, 

Note :

To do simplification (addition and subtraction) of algebraic expression, we can combine only like terms, not unlike terms. 

Examples

Example 1 :

Subtract 5xy from 8xy. 

Solution : 

Example 2 :

Subtract (3c + 7d²)  from  (5c - d²).

Solution : 

Alternatively, this can also be done as :

(5c - d²) - (3c + 7d²)  =  5c - d²  -  3c - 7d² 

=  (5c - 3c) + (-d² - 7d²)

=  2c  + (-8d²)

=  2c   - 8d²

Example 3 :

Subtract (2x² + 2y² - 6)  from  (3x² - 7y² + 9)

Solution : 

Example 4 :

Subtract (x³ + 4x² - x + 6) from (2x³ + 5x² - 2x + 7).

Solution : 

=  (2x³ + 5x² - 2x + 7) - (x³ + 4x² - x + 6)

=  2x³ + 5x² - 2x + 7 - x³ - 4x² + x - 6

=  2x³ - x³ + 5x² - 4x² - 2x + x + 7 - 6

=  x³ + x² - x + 1

Example 5 :

Subtract (2x³ + 7x² - 3x - 3) from (3x³ - 2x² - x + 4). 

Solution : 

=  (3x³ - 2x² - x + 4) - (2x³ + 7x² - 3x - 3)

=  3x³ - 2x² - x + 4 - 2x³ - 7x² + 3x + 3

=  3x³ - 2x³ - 2x² - 7x² - x + 3x + 4 + 3

=  x³ - 7x² + 2x + 7

Example 6 :

Subtract (3 + 3x - 7x⁵ - 5x⁶) from 2(x³ - x² + 6x - 2). 

Solution : 

=  2(x³ - x² + 6x - 2) + (3 + 3x - 7x⁵ - 5x⁶)

=  2x³ - 2x² + 12x - 4 - 3 - 3x + 7x⁵ + 5x⁶

=  5x⁶ + 7x⁵ + 2x³ - 2x² + 12x - 3x - 4 - 3

=  5x⁶ + 7x⁶ + 2x³ - 2x² + 9x - 7

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