An algebraic formula is a mathematical expression that uses algebraic symbols to represent relationships and calculations. These formulas involve variables, constants, and arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic formulas are fundamental tools in mathematics, enabling us to solve equations, describe patterns, and model real-world situations.
In this article, we have covered the algebraic expression definition, various algebraic formulas, related examples and others in detail.
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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 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 Formulas
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. Various algebra formulas used are added in the image below:
Algebraic Formulas Table
The below table consists of the important algebraic formulae.
| Algebraic Formulae | Expansion |
|---|---|
| (a + b)2 | = a2 + b2 + 2ab |
| (a - b)2 | = a2 + b2 - 2ab |
| a2 - b2 | = (a + b)(a - b) |
a2 + b2 a2 + b2 | = (a + b)2 - 2ab = (a - b)2 + 2ab |
| a3 + b3 | = (a + b)(a2 - ab + b2) = (a + b)3 - 3ab(a + b) |
| a3 - b3 | = (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 × n |
Uses of Algebraic Formulas
Various uses of Algebraic Formulas are:
- Solving Equations: Algebraic formulas help solve various types of equations, from linear to quadratic and beyond.
- Modeling Real-World Problems: Formulas can represent physical phenomena, financial calculations, engineering problems, and more.
- Simplifying Expressions: They enable simplification and manipulation of complex mathematical expressions.
- Predicting Outcomes: By plugging in known values, formulas can predict unknown quantities.
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Examples Related to Algebraic Formulas
Example 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
Example 2: Expand a3 - b3.
Solution:
= a3 - b3
= (a - b)(a2 + ab + b2)
Example 3: Multiply (x - y)(3x + 5y)
Solution:
= x(3x + 5y) - y(3x + 5y)
= 3x2 + 5xy - 3yx - 5y2
= 3x2 + 2xy - 5y2
Example 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