# Maths Tutors-How To Factorise a Polynomial

# Maths Tutors-How To Factorise a Polynomial-

## Factorisation-factorisation means writing a higher degree polynomial as a product of linear polynomials.

## Suppose we are given a quadratic polynomial and we are asked to factorise it then we have to try to write it as a product of two linear polynomials.

### If we are given a cubic polynomial and we are asked to factorise it then we have to try to write it as a product of three linear polynomials. this process continues for all higher degree polynomials.

### How to do the factorization- 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 factorisation 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 factorise it.

### Factorization of a binomial- we can have binomials of many types

##### 1.Binomial of degree two when both terms have same signs- these types of polynomials can’t be factorised, only a few can be factorised using perfect square identities.

##### 2.when both terms have opposite sign and power of variable is divisible by two-

##### these polynomials can easily be factorised by using a²-b²=(a+b)(a-b) identity

##### for example-1: 9x²-16y²

##### =(3x)²-(4y)²

##### = (3x-4y)(3x+4y)

##### we can also factorise 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 factorised by using a³-b³=(a-b)(a²+ab+b²) or a³+b³=(a+b)(a²-ab+b²)identity

##### for example-1: 64x³-27y³

##### =(4x)³-(3y)³

##### = (3x-4y)(9x²+12xy+16y²)

##### we can also factorise polynomials for degree 6, and degree 9 and much more in the same way

##### Factorization of a trinomial- A trinomial is usually a quadratic trinomial.This can be of two types:

##### 1.A perfect square quadratic trinomial can be solved using identity

##### (a+b)²=a²+2ab+b² or by (a-b)²=a²-2ab+b²

##### Example- 9x²-24x+16

##### =(3x)²-2(3x)(4)+(4)²

##### =(3x-4)²

##### 2. A generic (non-perfect square) quadratic trinomial then we factorise it using the middle term splitting method.

##### example: 9x²-25x+16

##### =9x²-(16x+9x)+16

##### =9x²-16x-9x+16

##### =x(9x-16)-1(9x-16)

##### =(9x-1)96x-1)

##### Factorization of cubic polynomials with four terms-these polynomials can be factorised by different ways.

##### 1. factorisation 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

**Example:**

**Example:**

##### Find the zeros of f(*x*) = 2x^{3} + 3*x*^{2} – 11*x* – 6

**Solution:**

**Solution:**

##### 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

##### 2x^{3} + 3*x*^{2} – 11*x* – 6

= (*x* – 2)(*ax*^{2} + *bx + c*)

= (*x* – 2)(2*x*^{2} + *bx + *3)

= (*x* – 2)(2*x*^{2} + 7*x + *3)

= (*x* – 2)(2*x* + 1)(*x* +3)

##### we have calculated a b and c by inspection or comparison method

##### 2.We can use binomial whole cube identity to factorise cubic polynomials that are perfect cubes in itself.

(a+b)³=a³+3a²b+3ab²+b³

##### this method is also used to factorise cubic polynomials with four terms but generally, we use it for 2 variables when two terms are perfect cubes and rest two are divisible by 3

##### Example:

##### 27x³+108x²y+144xy³+64y³

##### =(3x)³+3(3x)²(4y)+3(3x)(4y)+(4y)³

##### =(3x+4y)³

##### 3. Besides these methods we can use :

##### this method is also used to factorise 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)

##### 4 If we are ever asked to evaluate or factorise a³+b³+c³ we should first find the sum of a+b+c usually this sum is zero then we can use

##### a³+b³+c³=3abcd

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