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.

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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:
Find the zeros of f(x) = 2x3 + 3x2 – 11x – 6
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
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
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

click red text to dowlnload questions .pdf Factoring_Polynomials (1)

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