Factorization of Algebraic Expressions: Key Concepts
In this article, we will discuss some key concepts of factorization of algebraic expressions.
This involves expressing a number as the product of its prime factors.
For example, the prime factorization of 60 is:
60 = 2 x 2 x 3 x 5
The prime factors of 60 are 2, 2, 3 and 5.
Greatest Common Factor
Greatest common factor (GCF) of two or more expressions is the largest factor common to all the expressions. Finding the GCF helps simplify fractions and factor expressions.
For example, the GCF of 12 and 18 is 6.
GCF of 12 and 18 = 6
Quadratic expressions like x2 + 5x + 6 can be factorized into the product of two binomials, using various methods like taking out the common factor, grouping, completing the square, etc.
x2 + 5x + 6 = (x+2)(x+3)
Factoring quadratics allows us to solve quadratic equations or simplify expressions.
Difference of Squares
Expressions having the form a2 – b2 can be factorized as:
a2 – b2 = (a + b)(a – b)
9 x2 – 4 = (3x + 2)(3x – 2)
This factorization pattern is very useful in simplifying algebraic fractions.
Cubic polynomials like x3 + x2 + 3x + 1 can sometimes be factorized into binomials when they have a common factor.
x3+ 3x2 + 3x + 1 = x(x+1)(x+1)
Factorize completely: 24xy – 18x
24xy – 18x
= 6x(4y – 3)
Factorize: x^2 – 9
x2 – 9
= (x + 3)(x – 3)
Factorize: x3 – 8x
x3 – 8x
= x(x2 – 8)
= x(x – 2)(x + 4)
Why do we factorize algebraic expressions?
Factorization simplifies algebraic expressions, helps solve equations, and reveals the underlying structure of an expression.
What are the methods for factorizing quadratics?
Taking out the common factors, grouping, completing the square, and using the quadratic formula are some methods for factorizing quadratics.
When can cubics be factorized into binomials?
Cubic polynomials having a common factor can sometimes be factorized into binomials by taking the common factor outside.