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
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
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
2. A generic (non-perfect square) quadratic trinomial then we factorise it using the middle term splitting method.
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
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
2.We can use binomial whole cube identity to factorise cubic polynomials that are perfect cubes in itself.
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
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
click red text to dowlnload questions .pdf Factoring_Polynomials (1)