Factoring is essential to simplify the algebraic expressions. This topic is very useful for the students as without understanding the basic of factorial polynomials one would not be able to understand the later topics. Many students often confused between the factors and term, they think both are same, but no they are not. In any equation, we can add or subtract a term whereas factors need to be multiplied. You can understand them through their definitions.
Definition of factor and term:
A term can be added or subtract whereas when we multiply a term, then it is called as the factor. If the entire term in an equation is in product form, then it is a factored form of an equation. A term may or may not contain factors whereas a factor is made of a term. We can change the term into factor form by changing the expression into product form. However, remember that terms or value doesn’t get changed only its form gets manipulated.
For example- 3x(x+y) is an expression in factored form and3x+4y+z is an expression in non-factored form.
How to remove common factors?
Here you will learn how to get common factors and greatest common factor. To learn, let us take an example.
Example – Factor 2×2+4xy+8xy2
Solution- first we will take out the factors for each term
The factors of 2×2 are 1, 2, x, x2, 2x and 2x^2
The factors of 4xy are 1,2,4,x,y,2x,4x,y,2y,4y,2xy and so on
The factors of 8xy2 are1,2,4,8,x,y,2x,4x,8x,xy,xy^2 , 8xy^2 and so on
Take the greatest common factor from these. Out of these, the H.C.F is 2x. Now divide the whole term by this factor. You will get-
Check the correctness by solving the equation on another side, and if the L.H.S is equal to R.H.S, then it is true.
How to find greatest common factor using grouping method?
This method is not very popular, but it is quite useful when used.
Group the first and second terms, and you will get the two groups as shown here,
= (2x^2-x) + (8x-4)
=X (2x-1) +4(2x-1)
The common factor is 2x-1, the final factor form is 2x^2-x+8x-4= (2x-1) (x+4). Note that this method doesn’t work every time, and hence you cannot use it to solve every problem.
How to factor quadratic polynomials?
The quadratic equation is nothing but a second-degree polynomial, so the highest degree of expression will be 2. When finding factoring polynomials, you need to find a term which when multiply gives the third term of expression and when added gives the second term. If this condition is not satisfied then obtained factors are incorrect. It is easy to find the factors for some equations whereas some complex quadratic expressions do not give factors. In such situation, we use the quadratic formula.
How to find factor for equations with the higher degree?
To solve such problems, take the common factor from expression and use the above methods to get the factor.
Example – 3×5+3×4-9×3+18×2
Solution- the common factor is 3×2, so we can write the expression as 3×2(x^3+x^2-3x+6) to solve further use any of the methods above.