Mathematics is a subject with a vast scope of the study. It is divided into different branches like arithmetic, geometry, algebra, etc. As our level of study develops students get introduced to all these branches. Algebra is a branch of mathematics that is approached at the elementary level. In algebra, equations are expressed in terms of coefficients and variables where variables are unknown quantities. These expressions are solved with some given formulas to reach a solution.
Algebraic expression
An algebraic expression is an equation made up of terms consisting combination of coefficients, variables, and constants. The expressions are presented in the form of mathematical operations like addition, subtraction, multiplication, and division.
Components of an algebraic expression
- Coefficient: Coefficients are the fixed numerical values attached to the variable (unknown number). For example in the algebraic expression 5x2 + 2, 5 is the coefficient of x2.
- Variable: Variables are the unknown values that are present in an algebraic expression. For example in 4y – 1, y is the variable.
- Constant: Constants are the fixed numbers present in the expression. They are not combined with any variable. For example in the expression 5x + 2, +2 is the constant.
Algebraic formula
Algebraic formulas are the combination of numbers and letters to form an equation or formula. In an algebraic formula, numbers are fixed or constant with their known values. And, the letters represent the unknown values. The below table consists of the important algebraic formulae. In the table letters like a,b,c,m,n, etc represents the unknown quantities of the equation.
| Algebraic formulae |
|---|
| (a + b)2 = a2 + b2 + 2ab |
| (a – b)2 = a2 + b2 – 2ab |
| a2 – b2 = (a + b)(a – b) |
|
a2 + b2 = (a + b)2 – 2ab, Or a2 + b2 = (a – b)2 + 2ab |
| a + b = (a + b)(a2 – ab + b2) = (a + b)3 – 3ab(a + b) |
| a – b = (a – b)(a2 + ab + b2) = (a – b)3 + 3ab(a – b) |
| 2(a2 + b2) = (a + b)2 + (a – b)2 |
| (a + b)2 – (a – b)2 = 4ab |
| a4 + b4 = (a + b)(a – b)[(a + b)2 – 2ab] |
| (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca |
| (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca |
| a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca) |
| a4 + a2 + 1 = (a2 + a + 1)(a2 – a + 1) |
| a – b = (a4 + b4)(a2 + b2)(a + b)(a – b) |
| am × an = a(m + n) |
| (am)n = am x n |
Sample Problems
Question 1: Solve the equation (a + b + c)(a + b + c).
Solution:
Given expression, (a + b + c)(a + b + c),
now,
= (a + b + c)(a + b + c)
= (a + b + c)2
By the algebraic formula
= (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Question 2: Expand a3 – b3.
Solution:
= a3 – b3
= (a – b)(a2 + ab + b2)
Question 3: Multiply (x – y)(3x + 5y)
Solution:
= x(3x + 5y) – y(3x + 5y)
= 3x2 + 5xy – 3yx – 5y2
= 3x2 + 2xy – 5y2
Question 4: Simplify (y3 – 2y2 + 3y – 1)(3y5 – 7y3 + 2y2 – y + 4)
Solution:
= (y3 – 2y2 + 3y – 1)(3y5 – 7y3 + 2y2 – y + 4)
= 3y8 – 7y6 + 2y5 – y4 + 4y3 – 6y7 + 14y2 – 4y4 + 2y3 – 8y2 + 9y6 – 21y4 + 6y3 – 3y2 + 12y – 3y5 + 7y3 – 2y2 + y – 4
Simplifying the like terms,
= 3y8 – 6y7 + 2y6 + 13y5 – 26y4 + 19y3 – 13y2 + 13y – 4.