Algebraic Forms and Operations

Introduction

Al-Khwarizmi is widely known as one of the major figures in the history of algebra. In school mathematics, algebraic forms are used to represent unknown quantities with letters or variables.

Simple examples:

• the square of $3$ is $3^2 = 9$
• the square root of $9$ is $\sqrt{9} = 3$

A Problem Involving Algebraic Forms

A traveler rides for 3 hours at an average speed of $(2x-5)$ km/h, then continues for 4 hours at an average speed of $(4x+8)$ km/h. If the total distance traveled is 346 km, the value of $x$ can be found using algebraic expressions.

Definition of an Algebraic Form

An algebraic form is a mathematical expression that uses letters to represent unknown numbers.

Examples:

1. $5a + 6$
2. $2x - 1$
3. $x^2 + 2x + 3$

Elements of Algebraic Forms

Variable

A variable is a symbol used in place of a number whose value is not yet known. In the algebraic form $2x + y - 10$, the letters $x$ and $y$ are variables.

Coefficient

A coefficient is the numerical factor attached to a variable. In $3x - y + 7z + 8$:

• 3 is the coefficient of $x$
• -1 is the coefficient of $y$
• 7 is the coefficient of $z$

Constant

A constant is a term with no variable. In $2x^2 - 3x + 9$, the number 9 is the constant term.

Term

A term is a part of an algebraic expression separated by addition or subtraction. In $2x^2 - 5x + 12$, the terms are:

• $2x^2$
• $-5x$
• $12$

Like Terms

Like terms have the same variables raised to the same powers.

Examples:

• $3x^3$ and $4x^3$ are like terms
• $4x^2$ and $-7x^2$ are also like terms

Unlike Terms

Unlike terms do not have matching variables or exponents. For example, $3x$ and $3x^2$ are unlike terms.

Operations on Algebraic Forms

Addition and Subtraction

Addition and subtraction can only be done directly on like terms.

Examples:

1. $2x + 3x = 5x$

2. $8a - 6a = 2a$

3. Add $x + 7y - 4z$ and $-12x + 3y + 10z$

   $$\begin{aligned} (x + 7y - 4z) + (-12x + 3y + 10z) &= (x - 12x) + (7y + 3y) + (-4z + 10z)\\ &= -11x + 10y + 6z \end{aligned}$$

4. Subtract $6x^2 - 2x + 8$ from $4x^2 - 11x + 12$

   $$\begin{aligned} (4x^2 - 11x + 12) - (6x^2 - 2x + 8) &= 4x^2 - 11x + 12 - 6x^2 + 2x - 8\\ &= (4x^2 - 6x^2) + (-11x + 2x) + (12 - 8)\\ &= -2x^2 - 9x + 4 \end{aligned}$$

Multiplication and Division

Useful properties:

Examples:

1. $5(2x+y) = 10x + 5y$
2. $x(4x^2+9x+2) = 4x^3 + 9x^2 + 2x$
3. $(8x^2-4x):2x = 4x - 2$
4. $(12xy^2+6x^2y):3xy = 4y + 2x$

Evaluating Algebraic Forms

If variables are replaced with specific values, the value of the algebraic expression can be calculated.

Examples:

1. Find the value of $3x+4$ when $x=5$

   $$3x+4 = 3(5)+4 = 19$$

2. The product of two numbers is $36$. If one of the numbers is $-4$, find the other number.

   Let the unknown number be $x$.

   $$\begin{aligned} -4x &= 36\\ x &= \frac{36}{-4}\\ x &= -9 \end{aligned}$$

   So the other number is $-9$.

Powers in Algebraic Forms

Algebraic expressions may also contain powers, such as:

• $x^2$
• $3a^2b$
• $2x^3 - 5x^2 + x - 7$

An exponent shows how many times a variable is multiplied by itself. For example:

Closing

Understanding variables, coefficients, constants, terms, and the basic arithmetic operations on algebraic expressions is an important foundation for later mathematics topics.