**Operations on Polynomials:**

**1) Addition of Polynomials:**

Addition of Polynomials

1. Collect Like terms at one place

2. Add the numerical coefficients of like terms

3. Write the sum in both standard and simplest form

1. Add the following polynomials:

1. 2a + 3b and -4b + 5a 2. 6x + 2y -3z and 9z + 3y – 5x

1. We know what is meant by like terms. They are terms in which literal coefficients are same. So, to add like terms means to add the numerical coefficients of two or more polynomials which have same literal coefficients.

In 2a + 3b and -4b + 5a:

2a and 5a are like terms and 3b and -4b is another pair of like terms.

So, add them (the like terms):

2a + 5a = 7a

3b – 4b = -b

Now, 7a and –b are unlike terms which cannot be added like like terms.

So, the two unlike terms 7a and –b are written and the symbol

So, the sum of 2a + 3b and -4b + 5a is 7a – b

2. In 6x + 2y -3z and 9z + 3y – 5x

the like terms are

6x and -5x,

2y and 3y,

-3z and 9z

So, the sum of like terms is

6x – 5x = x

2y + 3y = 5y

-3z + 9z = 6z

Now write these sums connected by the addition sign

x + 5y + 6z

In the table below, the steps for subtraction of polynomials, the second basic operation on polynomials are clearly laid out:

Subtraction of Polynomials:

1. Subtract similar terms. To do this, change the algebraic sign of what is to be subtracted and add it to the other.

2. To subtract unlike terms, just write the operation sign – before what is to be subtracted

2. -4yz from –yz

1. Like in addition, we subtract one like term from another like term.

In 1, 9pq and 4pq are like terms. Also, 9pq needs to be subtracted from 4pq. To do this, subtract the numerical coefficients, as the polynomials are like terms.

Now note that 9 must be subtracted from 4, as that is what is asked of us. Therefore, write a

4 and -9

So 4pq – 9pq = -5pq

Since -4yz needs to be subtracted, change its sign and add it to –yz

-yz + 4yz ( on changing sign of -4yz, it becomes 4yz) = 3yz

5x

Now, to subtract 5x

2xy – (5x

Read the table below:

Multiplication of Polynomials:

• First multiply numerical coefficients and literal coefficients separately. Next, multiply these two products

• To multiply two polynomials when each one has more than one term: Multiply each term of one polynomial with each term of the other polynomial and write like terms together.

product of numerical coefficients 5 and 8 is 40 and product of literal coefficients p and q is pq.

Now, write the product of these two as: 40pq.

4x

Let us apply the 2nd rule in the above table:

(4x

4x

8x

Division of Polynomials:

1. To divide a monomial by another monomial, divide the numerical coefficients and the literal coefficients separately.

2. To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.

1. 50p

divide the numerical coefficients and write their quotient i.e, 50/5 = 10 now divide literal coefficients and write their quotient as

p

p

= p

Now, write the coefficients next to each other to denote their product50 p

Divide each in the polynomial 40a

Let us find the quotients separately as follows:

(40a

(55a

(35a

(70ab)/ (a

Now write the above four quotients next to each other, separated by the

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Note: Division of one polynomial containing more than one term by another similar polynomial is discussed in Intermediate Algebra. *