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# Operations With Polynomials (With Methods, Rules & Examples)

November 6, 2022

This post is also available in: हिन्दी (Hindi)

Polynomials are special types of algebraic expressions of the form $a_{n} x^{n} + a_{n – 1} x^{n – 1} + a_{n – 2} x^{n – 2} + … + a_{1} x + a_{0}$. They are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math, such as calculus. You can perform the four mathematical operations – addition, subtraction, multiplication, and division on polynomials.

Let’s understand the methods of operations with polynomials with examples.

## Operations with Polynomials

As you perform the four basic operations viz., addition, subtraction, multiplication, and division on real numbers, similarly you can perform these operations on polynomials.

The operations of addition and subtraction are different from the operations of multiplication and division in terms of types of terms in the polynomials.

You can add and/or subtract only the like terms of two polynomials. But in the case of multiplication and division, you can perform these operations with like terms as well as with unlike terms.

The addition of polynomials is simple. While adding polynomials, you simply add like terms. You can use columns to match the correct terms together in a complicated sum.

Remember these two rules while performing the addition of polynomials.

• Rule 1:  Always take like terms together while performing addition.
• Rule 2:  Signs of all the polynomials remain the same.

The steps used to perform the addition of two polynomials are

Step 1: Arrange the polynomial in standard form, i.e., in descending order the powers of the variable

Step 2:  Collect the like terms

Step 3: Add the like terms

Step 4: Arrange the sum of terms in descending order of the powers of the variable, i.e., the standard form of a polynomial

Maths can be really interesting for kids

### Examples

Let’s consider a few examples to understand the process of the addition of polynomials.

Ex 1: Add $5x^2 – 7x + 9$ and $3x^2 + 5x + 7$

The like terms in the two polynomials are

$5x^2$ and $3x^2$

$– 7x$ and $5x$

$9$ and $7$

Adding the like terms we get

$5x^2 + 3x^2 = 8x^2$

$– 7x + 5x = -2x$

$9 + 7 = 16$

Therefore, $\left(5x^2 – 7x + 9 \right) + \left(3x^2 + 5x + 7 \right) = 8x^2 – 2x + 16$

Alternatively

$\left(5x^2 – 7x + 9 \right) + \left(3x^2 + 5x + 7 \right)$

$=5x^2 – 7x + 9 + 3x^2 + 5x + 7$

$= \left(5x^2 + 3x^2 \right) + \left(- 7x + 5x \right) + \left(9 + 7 \right)$

$= 8x^2 – 2x + 16$

Ex 2: Add $5x^4 – 3x^3$, $4 + x^3$ and  $2x^4 – 7x^3$

The like terms in the two polynomials are

$5x^4$ and  $2x^4$

$-3x^3$, $x^3$ and  $- 7x^3$

$4$ is a lone term

Adding the like terms we get

$5x^4 + 2x^4 = 7x^4$

$-3x^3 + x^3 – 7x^3 = – 9x^3$

Therefore, $\left(5x^4 – 3x^3 \right)+ \left(4 + x^3 \right) + \left(2x^4 – 7x^3 \right) = 7x^4 – 9x^3 + 4$

Alternatively

$\left(5x^4 – 3x^3 \right) + \left(4 + x^3 \right) + \left(2x^4 – 7x^3 \right)$

$= 5x^4 – 3x^3 + 4 + x^3 + 2x^4 – 7x^3$

$= 5x^4 + 2x^4 – 3x^3 + x^3 – 7x^3 + 4$

$= 7x^4 – 9x^3 + 4$

## Subtraction of Polynomials

The subtraction of polynomials is as simple as the addition of polynomials. While subtracting polynomials, you simply subtract like terms. You can use columns to match the correct terms together in a complicated difference.

Remember these two rules while performing the subtraction of polynomials.

Rule 1: Always take like terms together while performing subtraction.

Rule 2: Signs of all the terms of the subtracting polynomial will change, $+$ changes to $-$, and $-$ changes to $+$.

The steps used to perform the subtracting of two polynomials are

Step 1: Arrange the polynomial in standard form i.e., in descending order the powers of the variable

Step 2:  Collect the like terms

Step 3: Enclose the part of the polynomial which to be deducted in parentheses with a negative $(-)$ sign prefixed. Then, remove the parentheses by changing the sign of each term of the polynomial expression.

Step 4: Subtract the like terms after altering the signs, if required

Step 5: Arrange the difference of terms in descending order of the powers of the variable, i.e., the standard form of a polynomial

### Examples

Ex 1: Subtract $5x^2 – 2x$ from $7x^2 + 8x$

The like terms in the two polynomials are

$5x^2$ and $7x^2$

$- 2x$ and $8x$

Subtracting the like terms we get

$7x^2 – 5x^2 = 2x^2$

$8x – (-2x) = 8x + 2x = 10x$

Therefore, $\left( 7x^2 + 8x \right) – (5x^2 – 2x) = 2x^2 + 10x$.

Alternatively

$\left(7x^2 + 8x \right) – \left(5x^2 – 2x \right)$

$7x^2 + 8x – 5x^2 + 2x$

Therefore, $\left( 7x^2 + 8x \right) – (5x^2 – 2x) = 2x^2 + 10x$.

Note: Change signs while opening the brackets.

$\left(7x^2 – 5x^2 \right) + \left(8x + 2x \right) = 2x^2 + 10x$.

Ex 2: Subtract $3x^2 + 2x – 5$ from $9x^2 – 2x$

The like terms in the two polynomials are

$3x^2$ and $9x^2$

$2x$ and $- 2x$

$- 5$ is a lone term

Subtracting the like terms we get

## FAQs

### What are the four basic mathematical operations that can be performed with polynomials?

With polynomials, you can perform addition, subtraction, multiplication, and division.

### Can you add or subtract unlike terms?

No, you cannot add or subtract unlike terms. Addition and/or subtraction are always performed on like terms.

### Can you multiply or divide unlike terms?

Yes, you can multiply or divide unlike terms. In fact, you can multiply or divide like terms as well as unlike terms.

### What operations result in a polynomial?

Whenever you add, subtract, or multiply two polynomials, you always get a polynomial. But this does not happen always with division. Division of two polynomials can result in a polynomial or can also result in a constant.

## Conclusion

With polynomials, you can perform the four basic mathematical operations, viz., addition, subtraction, multiplication, and division. In the case of addition and subtraction, only the like terms are added or subtracted, whereas, in the case of multiplication and division, you can multiply or divide like terms as well as unlike terms.

• cam says:

6x^(2+4) = 6x^(6), not to the 4th.

• Thank you for pointing out the error. I’ve rectified the error.

Regards,
Rajan Dobhal