Operations on Polynomials

Basic Algebra > Operations on Polynomials

Operations on Polynomials:

1) Addition of Polynomials:

In the following table, the steps for addition of polynomials, the first basic operation on polynomials are clearly laid out:

First Operation on 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

Solved Examples:

1. Add the following polynomials:
     1. 2a + 3b and -4b + 5a      2. 6x + 2y -3z and 9z + 3y – 5x

Solution:

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 ‘+’ is written to indicate the addition operation of polynomials in the given question
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 ‘+’ to indicate the sum of the two polynomials in the question (i.e. the addition operation on polynomials)

x + 5y + 6z

2) Subtraction of Polynomials:

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

Second Basic Operation on Polynomials:
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. Subtract: 1. 9pq from 4pq
2. -4yz from –yz

Solution:

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 sign for 9pq and then add the numerical coefficients

4 and -9
So 4pq – 9pq = -5pq

2. subtract -4yz from –yz

Solution:

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

3. Subtract:

5x2 + 6x from 2xy

Solution:
5x2 + 6x and 2xy are two unlike terms.

Now, to subtract 5x2 + 6x from 2xy, change the sign of what is to be subtracted i.e. 5x2 + 6x and add it to 2xy
2xy – (5x2 + 6x) = -5x2 + 2xy - 6x

3) Multiplication of Polynomials:

Multiplication of polynomials is the third important operation on polynomials.

Read the table below:

Third Operation on Polynomials
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.

Solved Examples:

Multiply the following polynomials:

1. 5p and 8q

Solution:

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.

2. 4x3 + 2 and 2x2 + 3x

Solution:

4x3 +2 and 2x2 + 3x
Let us apply the 2nd rule in the above table:
(4x3 +2)( 2x2 + 3x ) =
4x3.(2x2 + 3x) + 2(2x2 + 3x) = {Apply exponents rule: xm.xn = xm +n}
8x5 + 12x4 + 4x2 + 6x

1) Division of Polynomials:

Fourth Operation on Polynomials
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.

Solved Examples:

Divide the following polynomials
1. 50p4q6 by 5pq

Solution:

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

p4q6 by pq { recall exponent’s rule: {xm/xn = xm – n }
p4q6/ pq
= p4-1. q6-1 = p3.q5

Now, write the coefficients next to each other to denote their product50 p3.q5

2. Divide 40a5b4 + 55a3b5 + 35a3b4 + 70ab by a2b2

Solution:

Divide each in the polynomial 40a5b4 + 55a3b5 + 35a3b4 + 70ab by a2b2

Let us find the quotients separately as follows:

(40a5b4)/(a2b2) = 40a5 – 2 . b4 – 2 = 40a3b2
(55a3b5)/ (a2b2) = 55ab3
(35a3b4)/ (a2b2) = 35ab2
(70ab)/ (a2b2) = 70/(ab)

Now write the above four quotients next to each other, separated by the + sign to indicate their addition

40a3b2 + 55ab3 + 35ab2+ 70/ (ab)

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