**Operations with polynomials : **

A polynomial is an expression composed of coefficients and variables under addition, subtraction and multiplication and exponents on those variables must be non-negative integers.

Before going to learn how to add, subtract, multiply or divide polynomial we must know the terms

- Coefficients
- like terms

**What is coefficients ?**

A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.

For example, in -4x the coefficient of x is -4.

**What are like terms ?**

Like terms are terms that contain the same variables raised to the same power. Only the numerical coefficients are different. In an expression, only like terms can be combined.

For example, 5a, -3a, 2ab

in the above three terms 5a and -3a are like terms, because they are having same variable, but 2ab is not like term of 5a or -3a, because it has a different variable "ab" other than "a".

**Operations with polynomials :**

- Adding polynomials
- Subtracting polynomials
- Multiplying polynomials
- Dividing polynomials

Let us discuss the above topics one by one with detailed examples.Operations with polynomials

Adding polynomials is nothing but combining the like terms.We follow two different methods to add two or more polynomials

- Horizontal method
- Vertical method

In both ways we will get the same answer.

**Example 1 :**

Add : (3 x³- 5 x²+ 2 x - 7) and (4 x² + x - 8)

**Solution :**

Here we give step by step explanation for adding the above two polynomials using horizontal method.

**Step 1 :**

Before going to add two polynomials, first we have to arrange the given polynomials one by one from highest power to lowest power.

(3x³-5x²+ 2x-7) and (4x²+x-8)

The two given polynomials are already in the arranged form.So we can leave it as it is.

**Step 2 :**

Now we have to write the like terms together starting from the highest power to lowest power.

= 3 x³-5 x²+ 4 x²+ 2 x + x - 7 - 8

**Step 3:**

Combine the like terms (Add or subtract) based on the signs of those terms.

= 3 x³- x²+ 3 x - 15

We have to line up the polynomials from highest power to lowest power one by one. If we have any missing term in the given polynomials then we have to put 0 instead of the missing term.

Subtracting polynomials is also nothing but combining the like terms.We follow two different methods to subtract two polynomials

- Horizontal method
- Vertical method

In both ways we will get the same answer.

**Example 2 :**

Subtract the following polynomials:

(2 x³ - 2 x² + 4 x - 3)- (x³ + x² - 5 x + 4)

**Solution :**

**Step 1:**

In the first step we are going to multiply the negative with inner terms.

= 2 x³ -2 x² + 4 x - 3 - x³-x²+ 5 x - 4

**Step 2:**

In the second step we have to combine the like terms

= 2 x³ - x³ - 2 x²- x² + 4 x + 5 x - 3 - 4

**Step 3:**

After combining the like terms we will get the answer

= x³ - 3 x² + 9x - 7

**Subtracting polynomials in vertical method :**

We have to write the polynomials from highest power to lowest power one by one. If we have any missing term then we have to put 0 instead of that.

Change the signs of second polynomial from positive to negative and negative to positive. Hereafter we should not consider the original sign, we have to consider the changed sign.

According to the signs we have to combine the terms.

For more examples please visit "Adding and subtracting polynomials".

We will multiply two or more polynomials in the following order.

(1) Symbol

(2) Number

(3) Variable

Let us see how it works

Multiply ( 5 x² ) and (-2 x³)

= ( 5 x² ) **x** (-2 x³)

= 10 x⁵

**Example 3 :**

Multiply (2 a² + 5 a - 1) x (8 a² - 3 a + 5)

**Solution :**

= (2 a² + 5 a - 1) x (8 a² - 3 a + 5)

To multiply these trinomials we have to distribute 2 a² with (8 a² - 3 a + 5), distribute 5 a with (8 a² - 3 a + 5) and -1 with (8 a² - 3 a + 5).

= 16a⁴-6 a³+10 a²+40 a³-15 a²+25 a-8 a²+3 a-5

Now we have to combine the like terms

= 16a⁴-6a³+40a³+10a²-15a²-8a²+25a+3a-5

= 16a⁴ + 34 a³ -13 a²+ 28 a - 5

For more examples please visit "Multiplying polynomials".

The division of polynomials can be expressed by division algorithm.

**Example 4 :**

Find the quotient and remainder when 10 - 4x + 3x² is divided by x - 2

**Solution :**

To divide the given polynomial by other polynomial, first we have to write each polynomial is descending order.

Let f(x) = 3x² - 4x + 10 and g(x) = x - 2

Quotient = 3x + 2

Remainder = 14

After having gone through the stuff given above, we hope that the students would have understood "Operations with polynomials".

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