Polynomials are algebraic expressions that consist of variables with exponents, coefficients, and constants that are combined via elementary mathematical operations like addition, subtraction, and multiplication. The word “Polynomial” is made up of two Greek terms – “poly” meaning “many” and “nomial” meaning “terms”. Put together it means many terms. Although a polynomial can have any number of terms, it cannot be infinite.
Terms in a Polynomial
The parts in the polynomial separated by a plus sign “+” are called terms. Therefore every part of a polynomial can be said to be a term.
|x3– 2x2+ 3x+4||x2,-2x2,3x, and 4|
|3x+1||3x and 1|
Like and Unlike Terms
The terms having the same variable with the same exponential power are called like terms. Whereas, the polynomial terms having different variables or the same variable but with different exponential powers are called, unlike terms. Coefficients are not of significance here.
|2x, 3x, 5x and x||Like Terms|
|-5y2 and y2||Like Terms|
|2x3 and 3x5||Unlike Terms|
|x3 and y3||Unlike Terms|
Degree of polynomial
The highest exponential power of any variable present in any term of the algebraic expression is called the degree of the polynomial.
Types of Polynomials
Depending on the degree of the polynomial, they are divided into the following categories:
- Constant or Zero Polynomial: The polynomials with degree zero are called constant or zero polynomials. When the power of the variable becomes zero, then its value ultimately becomes 1. Such polynomials have only constant terms.
Example: 10, 94, 0, -6 etc.
- Linear Polynomials: The polynomials with the degree of one are called linear polynomials.
Example: x – 1, y + 1, a + 4 etc.
- Quadratic Polynomial: The polynomials with the degree of two are called quadratic polynomials.
Example: x2+ x, y2+ 1, a2+ 8 etc.
- Cubic Polynomial: The polynomials with the degree of three are called cubic polynomials.
Example: y3+ 8, x3– 27, 5+a3, x3+x2– 2 etc.
- Quartic Polynomial: The polynomials with the degree of four are called quartic polynomials.
Example: x4+x3-x2+ x+1, y4-y2+1, etc.
|Constant or Zero Polynomial||10||0|
|Quadratic Polynomial||x2+ y2 + 2||2|
|Cubic Polynomial||x3– 2x2+ 3x+4||3|
Classification of Polynomial
Depending on the terms of the polynomial, they are divided into the following categories:
- Monomial: The polynomials that consist of only one term.
Example: 4x, 3y, x2,y3,3a4 etc.
- Binomial: The polynomials that consist of two terms.
Example: x+1, x2– 1, y3+ 4, a+3, x2+ x, etc.
- Trinomial: The polynomials that consist of three terms.
Example: x2+ x+1, x2+ y2+2, y-3x+2, etc.
The process of obtaining the factors of a given value or an algebraic expression is called factorization. Before we understand the methods of factoring polynomials, let us first see what factors are. Factors are simply numbers multiplied together to obtain the original required number. Similarly, for the case of polynomials, the factors are also polynomials themselves multiplied together to generate the original polynomial.
Let us now have a look at the various techniques of factoring polynomials and most common factoring polynomials formulas.
How to factor polynomials, and what are the different factoring polynomials formulas?
Before we answer how to factor polynomials, let us first list down the various factoring polynomials formulas that we employ. There are five different factoring polynomials formulas as follows:
- Greatest Common Factor (GCF)
- Substitution Method
- Grouping Method
- Sum or difference in two cubes
- The difference in two squares method
Greatest Common Factor
In this method, we aim to find out the greatest common factor of the given polynomial to factorize it. It is simply a reverse procedure of the distributive law.
In the case of distributive law, we get:
p(q+r) = pq + pr
Whereas in the case of factorization, we invert the process
pq + pr = p(q+r)
Here p is the greatest common factor.
Also known as factoring by pair, the polynomial is distributed in pairs or grouped in pairs to find the zeros. The basic idea is to pair like terms together. So we can conveniently apply the distributive property to factorize it nicely.
Example: Factorize x2– 15x+50
Firstly find the two numbers that on being added give -15 as their sum and on multiplication give 50 as their product. The two numbers are -5 and -10 respectively as,
(-5) + (-10) = -15
(-5) x (-10) = 50
Therefore, we can rewrite the given polynomial as;
Taking x – 5 as common factor we get;
Hence, the factors are (x – 5) and (x – 10).
If the polynomial given is too complex, we can try substituting the complicated terms with a simpler term to solve. Thereby; making it much easier to factor out.
Example: Factorize (x – y)(x – y – 1) – 20
Let S = x-y. Now substitute S for x-y in the given expression.
(x – y)(x – y – 1) – 20 = (S)(S – 1) – 20
(S – 5)(S + 4)
Difference of Two Squares Identity
This technique applies to factorize the binomial expressions in the form of
x2– y2= (x – y)(x + y)
Example: Factorize (x+1)2– 9(x-2)2
(x+1)2– 9(x-2)2= (x+1)2– (3(x-2))2
How to factor cubic polynomials?
Students will often have to face the problem of how to factor cubic polynomials. The cubic polynomials are one of the most prominent polynomial forms asked in the questions to the students. The trick of factoring cubic polynomials lies in the use of identities. The sum and difference of the cube’s identity greatly help in factoring cubic polynomials. Let us now answer how to factor cubic polynomials in detail below.
Factoring Cubic Polynomials using the Sum and Difference of Cubes Identity
This technique applies to factorize the binomial expressions in the form of
x3+ y3= (x+y) (x2– xy+y2)
And, x3-y3=(x-y) (x2+xy+y2)
Example: Factorize 27x3 – y3
27x3– y3= (3x)3– y3
= (3x-y) ((3x)2+3xy+y2)
Example: Factorize 64x3+ 27y3
64x3+ 27y3= (4x)3+(3y)3
= (4x+3y) ((4x)2-12xy+(3y)2)
= (4x+3y) (16x2-12xy+9y2)
How to factor polynomials with four terms?
The next interesting question in our study is how to factor polynomials with four terms. The answer to how to factor polynomials with four terms is a simple one. We’ll have to follow the same method, as we have applied in the previous questions. Let us understand it with the help of an example.
Example: Factorize x3+x2-x-1
Break the polynomial into two parts as follows:
Now determine the highest common factor from both the parts and take it out of the brackets. As we can see from the first part, x2 is the greatest common factor. From the second part, we can take out -1 or the minus sign as common.
Again, regrouping the terms as the factors we’ll finally arrive at
Now that we’ve completely understood the various factoring polynomial formulas let us solve a few problems to gain more practice.
Question: Check if x+3 is a factor of x3+3x2+5x+15
Start by breaking the polynomial into two parts as follows:
Take out the greatest common factors from both the parts. For the first part x2 and the second part 5 is the greatest common factor.
Finally grouping the common terms we’ll arrive at the final solution.
Hence the answer is yes, (x+3) is a factor of x3+3x2+5x+15.
Question: Factorize x2+5x+6
Start by splitting the middle term into two numbers whose sum is 5 and product is 6. The two numbers are 2 and 3.
This is simply a polynomial with four terms. Break it into two parts.
Just like in the previous examples, take out the greatest common factors from the two parts. The common factors will be x and 3 for the first and second parts respectively.
All that’s left now is to group the common terms and reach the final solution.
Frequently Asked Questions
- What does it mean by factoring polynomials?
Factoring of a polynomial is essentially a method to break down the polynomial into a product of its factors.
- What are the methods of factoring a polynomial?
There are many methods to factorize a polynomial like greatest common factor, substitution method, grouping method, etc.
- What identities can we use to factor polynomials?
Identities like the difference of squares, and the sum & difference of cubes are very useful for factoring the polynomials.
- How do you factorize polynomials consisting of only two terms?
To factorize a polynomial with two terms we simply need to find the greatest common factor of both terms and take it out.
- What is the main trick to factoring polynomials with three terms?
To factorize polynomials with three terms, we need to split the middle term into two different terms whose sum is equal to the coefficient of the original middle term, and the product is equal to the third term in the original expression. After that, we can simply use the grouping method.