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.

Follow the below steps:

(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.