Operations With Polynomials (With Methods, Rules & Examples)

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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.

Addition of Polynomials

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

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

$9x^2 – $3x^2 = 6x^2$

$- 2x – 2x = -4x$

$0 – (-5) = 0 + 5 = 5$

Therefore, $\left( 9x^2 – 2x \right) – \left(3x^2 + 2x – 5 \right) = 6x^2 – 4x + 5$.

Alternatively

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

$= 9x^2 – 2x – 3x^2 – 2x + 5$

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

$6x^2 – 4x + 5$

Therefore, $\left( 9x^2 – 2x \right) – \left(3x^2 + 2x – 5 \right) = 6x^2 – 4x + 5$.

Multiplication of Polynomials

Multiplication of polynomials is a method for multiplying two or more polynomials together. The terms of the first polynomial are multiplied by the second polynomial to get the resultant polynomial. 

Based on the types of polynomials we use, there are different ways of multiplying them. The rules for the multiplication of polynomials are different for each type of polynomial. 

To multiply polynomials, the coefficient is multiplied with a coefficient, and the variable is multiplied with a variable.

In order to multiply any two polynomials the steps used are

Step 1: Multiply the coefficients

Step 2: Multiply the variables using exponent rules

Examples

Ex 1: Multiply $3x^2$ by $2x^4$

$3x^2 \times 2x^4 = (3 \times 2) \left(x^2 \times x^4 \right) = 6 x^{2 + 4} = 6x^6$.

Multiplying Polynomials with Different Variables

You can also multiply polynomials with different variables. The steps to multiply polynomials with different variables are:

Step 1: Multiply the coefficients

Step 2: Multiply the variables and use rules of exponents wherever necessary.

Examples

Ex 1: Multiply $2x^2$ with $4y$.

$2x^2 \times 4y = (2 \times 4) \left(x^2 \times y \right) = 8x^2y$.

Ex 2: Multiply $3a^2bc^3$ with $2abx$

$3a^2bc^3 \times 2abx = (3 \times 2) \left(a^2 \times a \right)(b \times b) c^3 x$

$= 6a^{2 + 1}b^{1 + 1} c^3 x = 6a^3b^2 c^3 x$

Multiplying Binomials

For multiplying binomials, we use the distributive property. Let’s multiply a binomial $(a + b)$ with another binomial $(c + d)$.

The step followed to multiply the two binomials are

Step 1: Write both the binomials together i.e., $(a + b)(c + d)$

Step 2: Out of the two brackets, keep one bracket constant, let’s say $(c + d)$

Step 3: Now multiply each and every term from the other bracket i.e., $(a + b)$ with $(c + d)$

Examples

Ex 1: Multiply $2x^2 + 5x$ by $3x – 2$

$\left(2x^2 + 5x \right)(3x – 2)$

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

$= 2x^2 \times 3x + 2x^2 \times (-2) + 5x \times 3x + 5x \times (-2)$

$= 6x^3 – 4x^2 + 15x^2 – 10x = 6x^3 + 11x^2 – 10x$

Division of Polynomials

Dividing polynomials is an arithmetic operation where we divide a polynomial by another polynomial, generally with a lesser degree as compared to the dividend. The division of two polynomials may or may not result in a polynomial. 

Similar to multiplication, in the division also the coefficient of one polynomial is divided by the coefficient of another polynomial, and the variable of one polynomial is divided by the variable of another polynomial.

In order to divide any two polynomials the steps used are

Step 1: Divide the coefficients

Step 2: Divide the variables using exponent rules. While dividing the variables, the rules of exponents are kept in mind.

Examples

Ex 1: Divide $12x^{3}$ by $2x$

$12x^{3} \div 2x = \frac {12x^3}{2x} = \frac{12}{2} \times \frac{x^3}{x} = 6x^{3 – 1} = 6x^2$

Ex 2: Divide $15x^3 + 5x^2 + 2x$ by $5x$

$\left(15x^3 + 5x^2 + 2x + 7 \right) \div 5x = \frac{15x^3 + 5x^2 + 2x + 7}{5x} = \frac{15x^3}{5x} + \frac{5x^2}{5x} + \frac{2x}{5x} = 3x^2 + x + \frac{2}{5}$

Practice Problems

1. Add the following polynomials
   • $3x^3 + 2x^2 – 5x + 7$, $-x^2 + 9$
   • $8x^2 – x  + 2$, $4x^3 + 8x – 9$, $2x + 5$
2. Subtract the following polynomials
   • $7x^3 – 2x^2 + 8x + 7$ from $12x^3 – 5x^2 – 8x + 19$
   • $12x^3 + 12x + 8$ from $8x^3 + 6x + 4$
3. Multiply the following polynomials
   • $5abx^3$ and $12a^2bx^2$
   • $-15x^2 + y^2$ and $2x^2 – 3y^2$
4. Divide the following polynomials
   • $12x^3 – 6x^2 + 3x$ by $3x$
   • $-15x^4 + 12x^3 + 16x^2$ by $4x^2$

FAQs

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.

Recommended Reading

• Factorization of Polynomials – Methods & Examples
• Factor Theorem of Polynomials – Definition, Proof & Examples
• Remainder Theorem of Polynomials – Definition, Proof & Examples
• What are Polynomials? (Definition, Types & Examples)
• Nature of Roots of Quadratic Equation(With Methods & Examples)
• Solving Quadratic Equations – Formulas, Tricks & Examples
• Quadratic Equation Definition (With Different Forms & Examples)
• Pair of Linear Equations in Two Variables(With Methods & Examples)
• Linear Equations in Two Variables – Definition, Types, and Graphs
• Linear Equations in One Variable – Graph & Method of Solving
• What are Algebraic Identities(With Definition, Types & Derivations)
• What is the Meaning of Equation – Definition, Types & Examples
• Division of Algebraic Expressions(With Methods & Examples)
• Multiplication of Algebraic Expressions(With Methods & Examples)
• Subtraction of Algebraic Expressions(With Methods & Examples)
• Addition of Algebraic Expressions(With Methods & Examples)
• What is Algebraic Expression(Definition, Formulas & Examples)
• What is Algebra – Definition, Basics & Examples
• What is Pattern in Math (Definition, Types & Examples)