Factorization of algebraic expressions involves finding two or more expressions that multiply together to give the original expression. It's like reversing multiplication. This process helps simplify and understand complex algebraic formulas. We'll also cover common algebraic identities that aid in this factorization process.
Introduction to Factors and Algebraic Expressions
Factors are numbers or quantities that, when multiplied together, produce a given number or expression. For example, the factors of 12 include 1, 2, 3, 4, 6, and 12.
In terms of prime factors, 12 can be expressed as 2² × 3¹.
Similarly, algebraic expressions involve variables, constants, and operators. They consist of terms separated by addition or subtraction. For instance, the expression 3xyz - 16x² - yz has three terms: 3xyz, -16x², and -yz.
Also Check: Factoring Polynomials
Algebraic Expressions Method
Factoring algebraic expressions involves several common methods. These methods include:
- Factorization using common faces: Identifying and factoring out the highest common factors from terms in the expression.
- Factorization by regrouping terms: Rearranging and regrouping terms within the expression to facilitate factoring.
- Factorization using identities: Applying algebraic identities to simplify and factorize expressions.
Factorization using common faces
To factorize an algebraic expression, we reverse the process of expanding it. This involves identifying common factors among terms and grouping them accordingly.
For example, consider the expression -3y2 + 18y:
We can factor out -3y: -3y2 + 18y = -3y(y - 6).
In this way, factoring simplifies and breaks down complex expressions into simpler components.
Also Check: Factors of 215
Factorization by regrouping terms
Factorization by regrouping terms involves rearranging algebraic expressions to identify common factors among the terms. For instance, consider the expression 12a + n - na - 12. Although not all terms share a common factor initially, we can regroup them based on shared factors:
The terms can be rearranged as shown:
12a + n - na - 12 = 12a - 12 + n - na
12a - 12 + n - an = 12(a - 1) - n(a - 1)
12a - 12 + n - an = (12 - n)(a - 1)
We can simplify algebraic expressions by grouping terms.
Also Check: Factors of 144
Factoring Expression using Standard identities
An identity in math is an equality that holds true for all values of variables. For instance:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
a2 - b2 = (a - b)(a + b)
These equations, known as identities, remain valid regardless of the values substituted for a and b.
Factorization of algebraic expressions is a key concept covered in Chapter 14 on factorization. We previously studied factorization of numbers, and now we explore how to factorize algebraic expressions into products of their factors.