Factorization
1 Definition
- Factorization is the breaking down of a complex mathematical expression into a product of simpler terms known as factors.
- The product of these factors, when multiplied, is equal to the original expression.
2 Factorization of polynomials
2.1 Polynomials
- Polynomial expressions contain different terms and can take the form: \[
ax^n+bx^{n-1}+cx^{n-2}+...+k
\]
\(a, b, c, ..., k\) are constants and \(n\) is a positive integer.
\(a, b, c, ...\) are referred to as the coefficients, while \(k\) is referred to as the constant term.
The constant term (\(k\)) can be considered as \(kx^0\).
The highest power (\(n\)) in the polynomial is known as the degree of the polynomial.
Based on the degree, the polynomial can be classified as:
Cubic polynomial, if the degree is three.
Quadratic polynomial, if the degree is two.
Linear polynomial, if the degree is one.
2.2 Methods of factoring polynomials
2.2.1 Greatest common factor
Factor \(x^2+3x\)
\[ x^2+3x= \]
\[x(x+3)\]
Factor \(4y^2+8y\)
\[ 4y^2+8y= \]
\[ 4y(y+2) \]
The common factor \(4y\) was chosen instead of \(2y\) because it is the greatest common factor.
2.2.2 The difference of two squares
- A binomial with the form \(x^2 - y^2\) can be factored to \((x + y)(x - y)\)
Factorize \(x^2-9\)
\[ x^2-9 = \]
\[ (x+3)(x-3) \]
Factor \(16y^2-49\)
\[ 16y^2-49= \]
\[ (4y)^2-(7)^2= \]
\[ (4y+7)(4y-7) \]
2.2.3 Quadratic expression
It is a polynomial of degree 2.
Its form is \(ax^2+bx+c\), where \(a\neq 0\).
It can be factored into two binomials.
If the coefficient \(a = 1\):
- Find two numbers that multiply to give the constant \((c)\) and their sum gives the coefficient \((b)\).
If the coefficient \(a \ne 1\):
Calculate \(a\times c\).
Find two numbers (e.g., \(b_1\) and \(b_2\)) that their product is equal to \((a\times c)\) and their sum gives the coefficient \((b)\).
Replace the term \(bx\) with the terms \(b_1x\) and \(b_2x\).
Simplify by taking a common factor of every two terms.
Simplify further by taking one parenthesis as a common factor.
Factor \(x^2-3x-10\)
\(a=1,\ b=-3,\ c=-10\)
The product of \(-5\) and \(2\) equals \(-10\ (\text {the constant}\ c)\) and their sum equals \(-3\ (\text {the coefficient}\ b)\), so this expression can be factored to:
\[ (x-5)(x+2) \]
Factor \(2x^2-7x-15\)
\(a=2,\ b=-7,\ c=-15\)
\(a\times c=2\times -15=-30\)
The product of \(-10\) and \(3\) is equal to \(-30\ (a\times c)\) and their sum gives \(-7\ (\text {the coefficient}\ b)\), so \(b_1\) and \(b_2\) are \(-10\) and \(3\).
The above expression becomes:
\[ 2x^2-10x+3x-15 = \]
\[ (2x^2-10x)+(3x-15) = \]
\[ 2x(x-5)+3(x-5) = \]
- Simplify further by taking \((x-5)\) as a common factor, so the expression can be factored to:
\[ (2x+3)(x-5) \]
2.2.4 Cubic factorization
2.2.4.1 Cubic binomial
Use sum or difference of cubes:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
2.2.4.2 Long or Synthetic division for cubic polynomial
If one root of the cubic polynomial (e.g., \(x = r\)) is known, long or synthetic division can be used to divide the cubic polynomial by \(x - r\).
The quotient will be a quadratic polynomial, which can then be factored further.
These methods are not covered here, but for technical details refer to Math is Fun for long division or BYJU’S for synthetic division.
3 References
Basic Algebra. Help Engineers Learn Mathematics (HELM) workbooks. Loughborough University. Retrieved August 13, 2024, from https://www.lboro.ac.uk/media/media/schoolanddepartments/mlsc/downloads/Basic%20Algebra.pdf
Factorisation. Byju’s. Retrieved August 13, 2024, from https://byjus.com/maths/factorisation/
Factoring. In Intermediate Algebra 1e (OpenStax). LibreTexts. Retrieved August 13, 2024, from https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/06%3A_Factoring