Polynomials: Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression that is composed of variables, constants, and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication, and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial, and trinomial. Examples of constants, variables, and exponents are as follows:

• Constants. Example: 1, 2, 3, etc.
• Variables. Example: g, h, x, y, etc.
• Exponents: Example: 5 in x⁵ etc.

Monomial: A monomial is an algebraic expression that has only one term. Examples are 3x, 5, x².

Binomial: A binomial is a polynomial or algebraic expression, which has a maximum of two non-zero terms. It consists of only two variables. Examples are 2x²+y, 10p+7q², a+b, 2x²y²+9, which are all binomials having two variables.

Trinomial: A trinomial is a polynomial or algebraic expression, which has a maximum of three non-zero terms. It consists of only three variables. Examples are 2x²y²+9+z, r+10p+7q², a+b+c, which are all trinomials having three variables.

Factoring polynomials is the reverse procedure of the multiplication of factors of polynomials. An expression of the form ax^n + bx^n-1 +kcx^n-2 + ….+kx+ l, where each variable has a constant accompanying it as its coefficient is called a polynomial of degree ’n’ in variable x. Thus, a polynomial is an expression in which a combination of a constant and a variable is separated by an addition or a subtraction sign.

Zeroes of polynomials, when represented in the form of another linear polynomial are known as factors of polynomials. After factorization of a given polynomial, if we divide the polynomial with any of its factors, the remainder will be zero. Also, in this process, we factor the polynomial by finding its greatest common factor. Now let us see how to factorize polynomials here with the help of few examples.

Factorization of Polynomials: The process of finding factors of a given value or mathematical expression is called factorization. Factors are the integers that are multiplied to produce an original number. For example, the factors of 18 are 2, 3, 6, 9, and 18, such as;

18 = 2 *9

18 = 2*3*3

18 = 3*6

Similarly, in the case of polynomials, the factors are the polynomials which are multiplied to produce the original polynomial. For example, the factors of x² + 5x + 6 are (x +2) and (x + 3). When we multiply both x +2 and x + 3, then the original polynomial is generated (Refer to Figure 1). After factorisation, we can also find the zeros of the polynomials. in this case, zeroes are x = -2 and x = -3.

Types of Factoring Polynomials: There are six different methods to factorizing polynomials. The six methods are as follows.

• Greatest Common Factor (GCF)
• Grouping Method
• Sum or difference in two cubes
• The difference in two squares
• General trinomials

Now in this article, let’s discuss the two basic methods which we use frequently to factorize the polynomial. Those two methods are the greatest common factor method and the grouping method. Apart from these methods, we can factorize the polynomials by the use of general algebraic identities. Similarly, if the polynomial is of a quadratic expression, we can use the quadratic equation to find the roots/factors of a given expression. The formula to find the factors of the quadratic expression (ax² + bx + c) is as shown in the figure below (Refer to Figure 2 for the quadratic formula).

Greatest Common Factor Method: In this method, identifying the greatest common factor is the only main step required in factoring a polynomial. This process is nothing but a type of reverse procedure of distributive law, such as;

p( q + r) = pq + pr

But in this case of factorization, it is just an inverse process as given below;

pq + pr = p( q + r)

where p is the greatest common factor.

Refer to Figure 3 given below which explains the GCF method with an example.

Factoring Polynomials by Grouping: The second technique of factoring is called grouping. This method is used when there is no factor common to all the terms of a polynomial, but there will be factors common to some of the terms. In other words, this method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros. Let us take an example.

Example: Factorise the polynomial x² -8x + 15.

Find the two numbers which when added together give a -8 and which when multiplied give a 15.

So, -3 and -5 are the two numbers, such that;

(-3) + (-5) = -8

(-3) * (-5) = 15

Hence, we can rewrite the polynomial as;

x² -3x -5x + 15

And then, group the first two terms which have the common factor ‘x’ and the next two terms which have the common factor of ‘-5’ as shown below,

x(x -3) -5(x-3)

Now the polynomial which had four terms has been reduced to a polynomial with just two terms. From these two terms, take out the common factor, which in this case is (x -3) as;

(x -3) (x-5)

Hence, the factors of x² -8x + 15 are (x-3) and (x-5) (Refer to Figure 4 for solution).

Sum or difference in two cubes: In this technique, the difference in two cubes means that there will be two terms and each will contain perfect cubes and the sign between the two terms will be negative. The sum of two cubes contains the plus sign between the perfect cube terms. The formula that helps in factoring polynomials with perfect cubes are as follows:

Sum: (x³+y³) = (x+y) (x² -xy +y²)

Difference: (x³-y³) = (x-y) (x² +xy +y²)

Example : Factorise the polynomial x³ + 64.

Here 64 is a perfect cube of 4. Hence we can rewrite 64 as 4³ as shown below.

x³ + 4³

To factor the above polynomial use the identity x³ + y³ = (x +y)(x² + xy + y²) as shown below (Refer to Figure 5 for solution).

x³ + 4³ = (x + 4) (x² +4x +4²)
= (x +4) (x² +4x +16).

When you have a difference of cubes, for example, x³ — 64 then you could use the same method to factorize. The only difference is that instead of using the identity sum of cubes you have to use the difference of cubes (Refer to Figure 6 for more details).

[ x³- y³ = (x- y)(x² +xy +y²) ].

The difference in two squares: A difference in two perfect squares defines that there should be two terms, where the sign between the two terms is a minus sign, and both the two terms contain perfect squares. The result after the factorization of the difference in two perfect squares should contain two binomial terms. One binomial term contains the sum of two terms whereas the other contains the difference of two terms.

We can say that, a² -b² = (a+b)(a-b) .

Example: Factorise the polynomial x⁴ -16.

Here x⁴ can be rewritten as (x²)² and 16 ad 4². Then, the polynomial can be written as :

x⁴ - 16 = (x²)² - 4²

Then, factor the above polynomial using the identity, a² -b² = (a+b)(a-b) as shown below;

(x²)² -4² = (x² +4)(x² - 4)

Here, x² + 4 is not further factor-able. But, x² -4 can be further factored using the same identity of difference of squares as shown below;

x² -4 = x² -2²

= (x + 2)(x -2).

so, x⁴ -16 = (x² + 4)(x + 2) (x -2) (Refer to Figure 7 for more detailed explanation with the help of an example).

Factoring using general trinomials: Whenever you have three terms in your polynomial (which is named as trinomial), and if it follows the identities of the sum of squares and difference of squares, then using the reverse procedure of multiplication, you can factor the given trinomial. (Refer to Figure 8 for step by step explanation).

The identity of sum of squares :

• (a + b)² = a² + 2ab + b²

The identity of difference of squares :

• (a — b)² = a² –2ab + b²

Example : Factorise the polynomial x² + 4x + 4.

Here, in the trinomial given, 4 and x are perfect squares and the middle term 4x is twice the product 2x. Hence, we can rewrite the given trinomial by using the identity of the sum of squares as;

x² + 4x + 4 = (x + 2)².

If the middle term was negative, then you could use the identity of the difference of squares to factor the trinomial.

More to come! Stay tuned, and thanks for reading. :)

All that matters for a person would be the person himself. Yet, each and everyone does something with a fear what others might think or say about them if they do so. Decisions regarding everything even what to opt for higher studies depends on the mindset of society. Why is it important? It’s not that we shouldn’t completely ignore the society. But why should one always consider what others might think?

Society contains variant mindset. Some may think positively or some may always consider negatively. What I’m saying is it’s not always necessary to think about all these stuffs. If one wish to do something which is good for them in life then let them do it. If your children want to do higher studies in photography , fashion designing or whatever it be then let them go for it instead of pushing them to become an engineer or a doctor for the so called status in society. Nowadays this so called status has ruined many life. If you ask many students why they chose engineering or medicine or whatever it might be their answer would be parents said to do so. By forcing them to study something which isn’t their dream or passion won’t mould good engineer or doctor.

Be it a man or a woman I don’t think one should sacrifice their dreams in fear of status or what the society might say about it. I personally feel that one should consider themselves in prior to others.

There is only life. Living life by enjoying each and every moment would give more satisfaction than living a life which has nothing to memorise other than regrets. Regretting in life is the most painful thing that a man can have as most of it cannot be fulfilled later in life.

So be brave , be bold and speak the truth and do all what you want which doesn’t harm yourself or others. Being You, being the real you is all that matters than being a common man. It’s time to think about being the real You. Better late than never. Be the real you in your life.