ALGEBRA FACTORING - FACTORS, PRIME POLYNOMIALS, FACTORING USING H.C.F., LINKS

Please study Polynomials before Algebra Factoring,
if you have not already done so.

It is a prerequisite here.

Factors or Divisors :

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

Example: We have (x + a)(x + b) = x² + x(a + b) + ab
Each of (x + a) and (x + b) are called the factors of x² + 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 polynomial as 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 reverse process of multiplication.

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, 2x² + 3y², 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.

Factorize 9x³y² + 3xy²z + 6x³y²z²

Solution:
Looking at all the terms, we can see the largest common term is 3xy²
∴ H.C.F. = 3xy²
Taking this out, the given expression becomes
3xy²(3x² + z + 2x²z²)

Thus the two factors of the given expression are
3xy² and (3x² + z + 2x²z²).

Thus, Algebra Factoring of 9x³y² + 3xy²z + 6x³y²z²
using H.C.F. gave factors as
3xy² and (3x² + z + 2x²z²). 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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Factorization by Rearrangement or Regrouping of terms in Algebra Factoring :

If we observe the terms of an algebraic expression
in the order they are given, they may not have a
common factor.

But by rearranging the terms in such a way
that each group of terms has a common factor,
some algebraic expressions can be factorised.

With more practice, you can easily regroup terms,
so that the given expression can be factorised.

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 the factors 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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Other Methods of Algebra Factoring :

For factoring of special products
using Algebra Formulas, go to

Factoring Special Products.

For factoring of Quadratic
Expressions, go to

Factoring Trinomials.

For factoring of Polynomials
using Factor Theorem, go to

Factoring Polynomials.