If two or more algebraic expressions are multiplied, their product is obtained.Each of the algebraic expressions which are multiplied to form the productare called the Factors (or Divisors) of the product.

Example: We have (x + a)(x + b) = x^{2} + x(a + b) + ab Each of (x + a) and (x + b) are called the factors of x^{2} + x(a + b) + ab

Factorization of Polynomials :

In Polynomials, we learnt how to multiply two Polynomials.

Can we reverse this process ? i.e. given a polynomial, can we find two or more polynomials whose product is equal to the given polynomial.

If we can, the process of doing this is called Factorization of Polynomials.

Thus,

Factorization of Polynomials means to express a polynomialas product of two or more polynomials. These two or more polynomials whose product is equal to the given polynomial are called the factors (divisors) of the polynomial. This is the reverseprocess of multiplication.

Algebra Factoring deals with Factorization of Polynomials.

Prime Polynomial :

A Polynomial is said to be prime if it is divisible by one and itself only.

Prime Polynomials are analogous to Prime Numbers in the set of Positive Integers. e.g. 3x + 4, 2x^{2} + 3y^{2}, 2 are prime Polynomials.

Finding the factors of a polynomial means finding all prime factors.

This is what we do in Algebra Factoring.

Common Factor :

A polynomial (constant polynomial or any degree polynomial) which divides all the terms of a polynomial is called a Common Factor of the terms of the Polynomial.

Highest Common Factor (H.C.F.) :

The largest polynomial (constant polynomial or any degree polynomial) which divides all the terms of a polynomial is called the Highest Common Factor (H.C.F.) of the terms of the Polynomial.

Factoring a Polynomial using H.C.F. :

A method of factorising a polynomial is to find the H.C.F. of all the terms of the polynomial and taking it out as a common factor.

Let us see some examples.

Example 1 of Algebra Factoring : Factoring using H.C.F.

Solution: Looking at all the terms, we can see the largest common term is 3xy^{2} ∴ H.C.F. = 3xy^{2} Taking this out, the given expression becomes 3xy^{2}(3x^{2} + z + 2x^{2}z^{2})

Thus the two factors of the given expression are 3xy^{2} and (3x^{2} + z + 2x^{2}z^{2}).

Thus, Algebra Factoring of 9x^{3}y^{2} + 3xy^{2}z + 6x^{3}y^{2}z^{2} using H.C.F. gave factors as 3xy^{2} and (3x^{2} + z + 2x^{2}z^{2}). Ans.

Example 2 of Algebra Factoring : Factoring using H.C.F.

Factorize (4a - 3b)(x - 2y) - (3a - b)(x - 2y)

Solution: Let A = (4a - 3b)(x - 2y) - (3a - b)(x - 2y) Here we can see (x - 2y) is the H.C.F. Taking this out, we get A = (x - 2y){(4a - 3b) - (3a - b)} = (x - 2y){4a - 3b - 3a + b)} = (x - 2y)(a - 2b)

Thus (x - 2y) and (a - 2b) are the two factors for the given expression.

Thus, Algebra Factoring of (4a - 3b)(x - 2y) - (3a - b)(x - 2y) using H.C.F.gave Factors as (x - 2y) and (a - 2b). Ans.

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Example 3 of Algebra Factoring: Regrouping of terms

Factorize 9x - 16y - xy + 144

Solution: Let A = 9x - 16y - xy + 144 Regrouping of terms, we have A = 9x + 144 - xy - 16y = 9(x + 16) - y(x + 16) = (x + 16)(9 - y) Ans.

Factorisation can also be done by another way of regrouping. A = 9x - xy + 144 - 16y = x(9 - y) + 16(9 - y) = (9 - y)(x + 16)

Thus Algebra Factoring of 9x - 16y - xy + 144 using Regrouping of terms gave thefactors as (9 - y) and (x + 16). Ans.

These ideas, taking out common factor and regrouping of terms are used in almost all problems of Algebra factoring.

We see more problems in different methods of Algebra factoring, the Links to which are given below.

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