Monomials and Polynomials of IX Grade

Hello friends, today we are going to learn about the Monomials and Polynomials. From its name the topic might look very difficult to a Grade IX student but it is as simple as addition and subtraction of numbers.

Let's start with the definition,

Monomial is the product of variables and the numbers(either positive or negative) with their exponents as whole numbers. The monomial terms does not contain any addition or subtraction terms in it.

In general the monomial can be written as xⁿ where n is a positive integer. If we take multiple variables like x, y and z then the monomial would be x^a y^b z^c with a, b and c as non negative integers.

Example- 3 a ² b ⁴, b d ³, –17 a b c are all monomials with no negative powers and no addition and subtraction terms.

As know we have the basic knowledge of what a monomial is so we can now perform different operations on it.

To start with the easy one we perform addition of the monomials.

Addition is done by taking out the common terms from the different monomials and writing the remaining ones in addition form.

Let's take different monomials ax ³ y ², 5b ³ x ³ y ² and c ⁵ x ³ y ². Now to perform addition on it we write these terms in addition form a x ³ y ² + 5 b ³ x ³ y ² + c ⁵x ³ y ² . Now from this addition equation we see that x ³ y ² are there in all the three terms so we take them common and write the remaining terms in addition form. So the result of this addition is ( a + 5b ³ + c ⁵ ) x ³ y ² . Now we can see that no more addition can be done, so this is our final answer of monomial addition.

ax ³ y ²+ 5 b ³ x ³ y ² + c ⁵ x ³ y ² = (a + 5b ³ + c ⁵) x ³ y ².

Subtraction is done in a similar way as the addition operation.

We take the same polynomials used above and perform the process of subtraction on them.

The polynomials are a x ³ y ², 5 b ³ x ³ y ² and c ⁵ x ³ y ². Now by writing them in subtraction form we get a x ³ y ²- 5 b ³ x ³ y ^{2 –} c ⁵ x ³ y ². We again see that the term x ³ y ² are common so we take them common and write the remaining terms in the subtraction format. So the result of this is ( a - 5b³ - c⁵ ) x ³ y ².

as no more operation could be done on it so this is the final answer of subtraction of monomial.

a x ³ y ²- 5 b ³ x ³ y ²- c ⁵ x ³ y ² = (a – 5b ³ – c ⁵) x ³ y ².

In the process of multiplication of two, three or multiple monomials, the numerical coefficients are just multiplied in the simpler manner and the exponents with common bases are just added and written .

Example – we take two monomials 5ax ³ z ⁸and –7a³ x ³ y ². Now to perform multiplication as stated above we just multiply the constant coefficients and the exponents for a,x and y are added and written.

The exponent of a becomes (1+3 =) 4, the exponent of x becomes (3+3 =) 6, the exponent of y becomes (0+2 =) 2 and the exponent of z becomes ( 8+0=) 8. So our final answer for the multiplication of these two terms is – 35a ⁴ x ⁶ y ² z ⁸.

5ax ³ z ⁸( –7 a ³ x ³ y ²) = –35 a ⁴ x ⁶ y ² z ⁸.

Just like addition and subtraction are similar to each other, in the same manner division and multiplication are also related to each other. In multiplication the exponents are added and the constant terms are multiplied, in a same way in division the constant terms are divided in simple way and the exponents gets subtracted.

Example- we take 35a⁴ x ³ z ⁹ and 7ax²z⁶ as our monomials to be divided. The first one is to be divided is called as the dividend and the one from which we are dividing is called as the divisor.

Dividend- 35 a ⁴ x ³ z ⁹.

Divisor- 7ax²z⁶.

Now as stated above we divide the constant coefficient in a simple manner and subtract the exponents with common base. The exponent of a becomes (4-1 =) 3, of x changes to (3-2 =) 1 and for z it changes to (9-6 =) 3. So our final answer is 5a³ x z ³.

35a ⁴ x ³ z ⁹: 7a x ² z ⁶ = 5 a ³ x z ³ .

Now we come over to polynomials.

Let's start with the definition,

When different monomials are added together then they form a Polynomial.

There are two types of polynomial - binomial and trinomial.

If two monomials are added then they are called as binomial. In general we write a binomial as x^a+y^b .

Examples - 3x+1, x² – 4x, 2x + y, or y – y².

If three monomials are added together then they are called as trinomial. In general a trinomial is written as x^a+y^b+z^c. Examples - x² + 2x + 1, 3x² - 4x + 10, 2x + 3y + 2.

If three or more than three monomials are added then they are represented in general as polynomials.

Examples - x² + 2x, 3x³ + x² + 5x + 6, 4x - 6y + 8.

Since now we have the basic knowledge of what a polynomial is we perform the basic operations on it also.

But first what we have to understand the concept of like and unlike terms.

Like terms are those terms which have the same variables and the exponents, there is no need for the constant terms to match. For example, 8xyz² and −5xyz² are like terms as they have same variables and powers.

Unlike terms are those which do not have the same variables or different exponents for same variables.

Example-5x³z⁸ and 7x²z⁶ are unlike terms.

To start with the simple one, we take addition.

In the process of addition we take different polynomials and the like terms get added while the unlike terms are left as they are.

Example- (y²- 3y+ 6) + (y- 3y²+ y³⁾

y²- 3y+ 6+ y- 3y²+ y³

Now we combine the like terms together,

y³ + y²- 3y²- 3y+ y+ 6,

Now we add the like terms and keep the unlike terms as they are

-y³- 2y²- 2y+ 6.

so this our final answer.

To perform Subtraction.

In subtraction also the like terms are subtracted while the unlike terms remain untouched.

For Example- (5x² + 2x +1) - ( 3x² – 4x –2 )

we remove the parenthesis,

5x² + 2x +1 - 3x² + 4x +2

and put the like terms together --

5x² - 3x² + 2x+ 4x+1 + 2,

and we get our final answer

2x²+ 6x +3.

Multiplication of polynomials is done by simply multiplying each term of one polynomial with each term of the other polynomial.

For example - ( p+ q+ r ) a = pa+ qa+ ra.

Now taking a complex one,

= ( x+ y+ z )( a+ b )

multiply each term of first polynomial with the each term of second polynomials.

= x( a+ b )+ y( a+ b ) + z( a+ b )

and now open the parenthesis and multiply to get the final answer.
= xa + xb + ya + yb + za + zb .

The process of division is also as simple as the other operations done on the polynomials.

In the division process we just separate the terms in the numerator and write the denominator in each one of them.

For Example – 2x+4 : 2

So separating the numerator we get,

2x/2 + 4/2 =

so the final answer becomes.

x+2.

Now we move forward to complex problems.

For Example - 21x ³ – 35x² : 7x.

We separate the numerator in the same manner as we have done above. So it becomes,

21x ³:7x – 35x²: 7x.

Now simply dividing this, we get

3x²- 35x.

So this is our final answer.

So now it is expected that with this basic knowledge about the monomial and the polynomial you would be able to perform the different operations on it.