How To Factorize a Polynomial
How to factor a polynomial
First of all, we need to understand the meaning of factorization. Factorization means writing a higher degree polynomial as a product of linear polynomials.
Suppose we have a quadratic polynomial and we want to factorize it then we have to try to write it as a product of two linear polynomials.
If we have a cubic polynomial and we want to factorize it then we have to try to write it as a product of three linear polynomials. This process continues for all higher degree polynomials.
Types of polynomial
There are many different types of polynomials classified on the basis of their degree and their number of terms, we have a different way of factorization for almost every type of polynomials
Factorization of a Monomial- Monomial is already a linear polynomial with degree one so we don’t need to factorize it.
how to factor a polynomial with two terms
A polynomial with two terms is called a binomial. we can have binomials of many types
Binomial of degree two
When both terms have the same signs- these types of polynomials can’t be factorized, only a few can be factorized using perfect square identities.
When both terms have opposite sign and power of the variable is divisible by two-
these polynomials can easily be factorized by using a²-b²=(a+b)(a-b) identity
we can also factorize polynomials for degree 4, degree 6, and degree 8 and much more in the same way
When both terms have opposite sign and power of variable is divisible by three- these polynomials can easily be factorized by using a³-b³=(a-b)(a²+ab+b²) or a³+b³=(a+b)(a²-ab+b²) identity
we can also factorize polynomials for degree 6, and degree 9 and much more in the same way
How to factor a polynomial with three terms
A polynomial with three terms is called a cubic polynomial. A trinomial is usually a quadratic trinomial. This can be of two types:
A perfect square quadratic trinomial can be solved using this identity
(a+b)²=a²+2ab+b² or by (a-b)²=a²-2ab+b²
A generic (non-perfect square) quadratic trinomial then we factorize it using the middle term splitting method.
Factorization of cubic polynomials with four terms-these polynomials can be factorized in different ways.
Factorization by using hit and trial method-
we use this method for cubic polynomials of 3 or 4 terms when we have only one variable in the polynomials. hit and trial is used when terms are usually in order
Find the zeros of f(x) = 2x3 + 3x2 – 11x – 6
We will find one solution to this polynomial by hit and trial method
Step 1: Use the factor to test the possible values by hit and trial.
f(1) = 2 + 3 – 11 – 6 ≠ 0
f(–1) = –2 + 3 + 11 – 6 ≠ 0
f(2) = 16 + 12 – 22 – 6 = 0
We find that the integer root is 2.
Step 2: Find the other roots either by inspection or by synthetic division. I am showing the inspection method here, you should try division method yourself
2x3 + 3x2 – 11x – 6
= (x – 2)(ax2 + bx + c)
= (x – 2)(2x2 + bx + 3)
= (x – 2)(2x2 + 7x + 3)
= (x – 2)(2x + 1)(x +3)
we have calculated a b and c by inspection or comparison method
We can use binomial whole cube identity to factorize cubic polynomials that are perfect cubes in itself.
We can use this method to factorize cubic polynomials with four terms also but generally, we use it for 2 variables when two terms are perfect cubes and rest two are divisible by 3
Besides these methods we can use :
We can also use this method to factorize cubic polynomials with 4 terms. generally, we use it for 2 or 3 variables when 3 terms are perfect cubes and 4th term is divisible by 3
a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab -bc -ca)
If we are ever asked to evaluate or factorize a³+b³+c³ we should first find the sum of a+b+c usually this sum is zero then we can use
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