Ans: Factoring of a polynomial function is one of the most important technique used in several mathematical applications like root finding, partial fraction method, stability analysis by pole zero calculation and more advanced applications in engineering and science. A polynomial is defined as a function of one or more variables containing higher powers of those variables only. For example, f(x) = x4-16 is a polynomial of degree 4 of variable x. factoring the polynomial is a process where the function is represented as a product of two or factors. It is similar to factoring of a number as a product of smaller numbers or primes, the only difference is that each factor is an expression of the variable instead of just a number. As an example the above function can be factored as (x2+4)(x+2)(x-2). This is called positive prime factoring that is the factoring process goes on until all the factors become positive primes. There are total six types of factoring which are Greatest Common factor method, Grouping, Difference in two squares, sum or the difference in two cubes, trinomials and general trinomials respectively.
Greatest common factor method: This is one of the most used factoring technique as this occurs in many other types of factoring method. In this method the largest common divisor of all the terms in a polynomial expression is taken common from the expression and represented as a factor. For example if (7x – 56) is required be factored then the GCF of the two terms 7x and 56 which is 7 is taken common from the two terms and then expressed as factors.
Hence, (7x-56) = 7(x-8)
Grouping: In many cases there is no whole factor common in all the terms of an expression but there is a common factor in some terms. In that case the terms with common factors are grouped together and then the factor is taken out from those term and finally the resultant expression is expressed as product of sum form.
Factorizing the expression 3ax + 6ay + 4x + 8y.
3ax + 6ay + 4x + 8y = 3a(x+2y) + 4(x+2y) = (x+2y)(3x+4)
Factorizing 12cx + 2y + 3x + 8cy = 12cx + 8cy + 3x + 2y = 4c*(3x + 2y) +1*(3x + 2y)
= (4c + 1)(3x+ 2y)
It is seen that the terms inside each of parentheses set are the same and thus this term becomes the greatest common factor.
Difference of two perfect squares: It is represented as a difference of two terms and magnitude of both terms are perfect squares. The factoring of this type of expression gives two binomials of which one contains the sum of two terms and the other term contains the difference of two terms.
As a generalized expression a2-b2=(a+b)(a-b)
Sum or difference in two perfect cubes: This is factorizing difference or sum of two cubical terms. The sum formula of two cubes is given by,
The difference of two cubes is given by,
The SOAP acronym is commonly used for remembering the arrangement of the signs.
Here, S represents that the sign between the two terms in the binomial expression in the factorized form will be same as the sign which is given in the problem.
By the difference formula of the cubes it is known that in the answer term there will be the difference between the cube roots of the two terms. Next, the other factor in the answer will contain two square terms of the two terms in the expression and the multiplicand of the two terms. Now, 64 is a perfect cube of 4, hence, the answer becomes
Trinomial: Before factoring the trinomial expression the terms of the expression must be arranged in a descending order. In most cases the trinomials are product of two binomial expressions.
Factorization of the expression X2+7X+12
The terms are already in descending order. Then the following process is followed.
Factorization of -16X-2X+3X2
At first the terms are arranged in descending order.
General trinomial: In a general trinomial the coefficient of the first term cannot be factored from the GCF and hence trial and error method is used to determine the factors of the trinomial expression.
At first the negative sign is taken common from all terms.
In the simplest case the most common technique is to take common from series of terms (GCF method) and if there is no common in all terms then the later methods are applied. This GCF method is also known as un-distribution as the distribution law is applied in reverse.
Special case: trinomial of perfect square
A trinomial of perfect square is an expression which can be expressed as square of some expression or basically the expression can factored in two same factors only. In general the perfect square trinomial is expressed as
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