Factoring

When solving higher degree equations, factoring is a useful tool for simplifying many algebraic expressions. Without understanding factoring, it is extremely difficult to progress in algebra beyond this point.

It has been stressed in earlier chapters that terms and factors should be distinguished. Addition and subtraction of terms and multiplication of factors should be remembered. Here are three definitions that are important to understand.

Sums or differences are indicated in terms. Indicated products contain factors.

Factors

Numbers have factors:

$7\times2=14$

7 and 2 are factors.

Factors in Algebra

Factors are commonly used in algebra.

Example:

x^2-x-6

=(x+2)(x-3)

(x+2)(x-3) are factors.

An important part of factoring (called factorizing in the UK) is to identify the factors that need to be considered:

A factor is an expression that can be obtained by multiplying two things together.

An expression is divided into simpler ones by multiplying them together.

A given expression is:

8x+6y+4z

=2(4x+3y+2z)

2(4x+3y+2z) are factors of 8x+6y+4z.

8x+6y+4z is factored in 2(4x+3y+2z).

Two criteria must be met for factoring to be correct:

The original expression must be obtained by multiplying the factored expression.
It is necessary to factor the expression completely.

Important points about factoring

Odd numbers are products of odd numbers.

Even numbers give an even product.

The product of an even number and an odd number is an even number.

The Sum of two numbers is even.

The Sum of two even numbers is even.

some formulas in factor form:

$a^2-b^2=(a+b)(a-b)$ .

$a^2+2ab+b^2=(a+b)(a+b)$ .

$a^2-2ab+b^2=(a-b)(a-b)$ .

$a^3+b^3=(a+b)(a^2-ab+b^2)$ .

$a^3-b^3=(a-b)(a^2+ab+b^2)$ .

Factors

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