# Subtracting Polynomials || How to subtract polynomials?

In this post we will learn methods for subtracting polynomials with solved examples.

## Methods of Subtracting Polynomials

Here we will learn two methods:

(a) Horizontal Polynomial Subtraction
(b) Vertical Polynomial Subtraction

Both the methods follow some given basic rules:

(1) Subtraction can only be done with Like Terms.
Any entity with same variables is known as like terms.

(2) While subtraction only coefficient of entity change, the variable remains the same.

### Horizontal Polynomial Subtraction

Suppose we are subtracting expression A with B (i.e. A – B)
For correct subtraction, follow the below steps:

(a) Enclose the expression B in a bracket, keeping – sign outward.

(b) Remove the bracket and change the sign of all the entities in expression B.

(c) Identify the like terms in A & B and complete the subtraction/addition as per given expression.

Let us understand the concept with examples.

#### Subtracting Polynomial Examples – Horizontal Subtraction

Example 01
A = 6x – y + 3
B = 2y – x
Subtract A – B

Solution
(i) Enclose expression B in bracket, keeping – sign outside.

6x – y + 3 – (2y – x)

(ii) Remove the bracket and change the sign of entities

6x – y + 3 – 2y + x

(iii) Identify the like terms and simplify

6x + x – y – 2y + 3

7x – 3y + 3

Note
While adding/subtracting like terms, only coefficient change, the variable remains the same.

Example 02
A = \mathtt{-7y^{2} +\ 2xy\ +\ 3x^{2}}
B = \mathtt{9x^{2} \ -6xy\ +\ 3y^{2}}

Subtract A – B

Solution
(i) Enclose B in bracket keeping -ve sign outside

\mathtt{-7y^{2} +\ 2xy\ +\ 3x^{2} \ -\ \left( 9x^{2} \ -6xy\ +\ 3y^{2}\right)}

(ii) Removing the bracket and changing the sign of each entity

\mathtt{-7y^{2} +\ 2xy\ +\ 3x^{2} \ -\ 9x^{2} \ +\ 6xy\ -\ 3y^{2}}

(iii) Find the like term and then simplify

\mathtt{-7y^{2} \ -\ 3y^{2} +\ 3x^{2} \ -\ 9x^{2} \ +\ 2xy\ +\ 6xy}\\\ \\ \mathtt{-10y^{2} -6x^{2} +8xy}

Example 03
A = \mathtt{2x^{3} \ +\ 4x^{2} +\ 7}
B = \mathtt{-5y^{3} \ +\ 11x^{2} \ -\ 10}

Subtract B – A

Solution
Note that we have to subtract expression B with A.
So we have to enclose bracket in expression A.

(i) Enclose A in bracket keeping -ve sign outside.

\mathtt{-5y^{3} +11x^{2} -10\ -\ \left( 2x^{3} +4x^{2} +7\right)}

(ii) Remove the bracket and change sign of all entities.

\mathtt{-5y^{3} +11x^{2} -10\ -\ 2x^{3} -4x^{2} -7}

(iii) Find the like terms and simplify

\mathtt{-5y^{3} -\ 2x^{3} +11x^{2} -4x^{2} -10-7}\\\ \\ \mathtt{-5y^{3} -\ 2x^{3} +7x^{2} \ -\ 17}

Example 04
A = \mathtt{\ x^{3} -9x^{2} -\ 1}
B = \mathtt{-6x^{2} -4}

Subtract A – B

Solution
(i) Enclose expression B in bracket keeping -ve sign outside

\mathtt{x^{3} -9x^{2} -\ 1\ -\ \left( -6x^{2} -4\right)}

(ii) Open the bracket and change the signs of expression B

\mathtt{x^{3} -9x^{2} -\ 1\ +\ 6x^{2} +4}

(iii) Find the like term and simplify

\mathtt{x^{3} -9x^{2} \ +\ 6x^{2} +4\ -\ 1}\\\ \\ \mathtt{x^{3} -3x^{2} +\ 3}

### Vertical Subtraction Polynomial

Suppose you have to subtract polynomial A – B.

(a) Arrange the like terms vertically in the same column

(b) For subtraction, change the sign of all entities in polynomial B

(c) Do the subtraction vertically and find the answer.

Note:
Due to subtraction sign in front of B, we have to change sign of all its entities.

Let us understand the concept with examples

#### Subtracting Polynomial Vertically Examples

Example 01
A = 4xy – 3x + 5y
B = 4y -2x + xy

Find A – B

Solution
(i) Arrange the like terms vertically

(ii) Remove the bracket and change the sign of expression B

(iii) Do the subtraction vertically

Hence, 3xy – x + y is the solution.

Example 02
A = \mathtt{x^{3} +2xy\ -\ 5x^{2}}
B = \mathtt{3xy\ -6x^{3} +\ 6}

Subtract A – B

Solution
(i) Arrange the like terms vertically

(ii) Remove the bracket & change signs of entity B

(iii) Subtract the entities vertically

Hence, \mathtt{7x^{3} -xy\ -\ 5x^{2} -6} is the solution.

Example 03
A = 2xy – x + 3y + 4
B = 6x + y

Subtract B – A

Solution
Here we are subtracting expression B with A.
So in vertical arrangement B will be in Top and A in bottom.

(i) Arrange the like terms vertically

(ii) Open the bracket and change the sign of expression A

Hence, 7x – 2y – 2xy – 4 is the solution

## Frequently asked questions – Algebraic Expression

(01) Can we subtract the unlike terms in polynomials?

NO!!
Unlike Terms are the entities with different variables.
It is not possible to subtract entities having different variables.

For Example;
Let 2x and 3xy are unlike terms.

If we subtract 2x – 3xy, the further subtraction is not possible.

(02) Difference between addition and subtraction of polynomial

One basic difference is that in subtraction we change the sign of subtracting polynomials.