Algebraic Expressions and Identities - Fundamental Concepts

Fundamental Concepts of Algebra

• Constants:

  • A symbol that has a fixed value is called a constant.
  • Examples of constants include: $7$, $-3$, $\frac{2}{5}$, $\frac{7}{2}$, $\sqrt{2}$, $2 - \sqrt{3}$, and $\pi$.

• Variables (Literals):

  • A symbol that can be given various numerical values is called a variable or literal.
  • Example: In the formula for the circumference of a circle, $C = 2\pi r$, where $C$ is the length of the circumference of the circle and $r$ is its radius:
  • $2$ and $\pi$ are constants.
  • $C$ and $r$ are variables (literals).

• Rules of Operations for Variables:

  • Since variables represent numbers, they obey all standard rules of operations of addition, subtraction, multiplication, and division of numbers. For any variables (literals) $x$, $y$, and $z$:
  • Commutative Law of Addition: $x + y = y + x$
  • Associative Law of Addition: $(x + y) + z = x + (y + z)$
  • Additive Identity: $x + 0 = x = 0 + x$
  • Additive Inverse: $x + (-x) = 0 = (-x) + x$
  • Commutative Law of Multiplication: $xy = yx$
  • Associative Law of Multiplication: $(xy)z = x(yz)$
  • Multiplicative Identity: $1x = x = x1$
  • Distributive Law: $x(y + z) = xy + xz$ and $(y + z)x = yx + zx$
  • Multiplication by Zero: $0x = 0 = x0$

• Powers of a Variable:

  • Let $x$ be any variable (literal):
  • $x \times x$ is written as $x^2$ and is read as $x$ square.
  • $x \times x \times x$ is written as $x^3$ and is read as $x$ cube.
  • General exponential rule: If $x$ is any literal and $n$ is a natural number, then $x^n = x \times x \times \dots \times x$ (multiplied $n$ times).
  • $x$ is called the base.
  • $n$ is called the exponent or index.
  • $x^n$ is the exponential form, read as $x$ raised to the power $n$, or $x$ to the power $n$, or simply $x$ power n$.\n * Particular rules:\n * x^1 = x\n * x^0 = 1\n\n# Algebraic Expressions and Terms\n\n* **Algebraic Expression**:\n * A collection of constants and literals (variables) connected by one or more of the operations of addition, subtraction, multiplication, and division is called an algebraic expression.\n\n* **Terms of an Algebraic Expression**:\n * The various parts of an algebraic expression separated by +$or$- signs are called terms of the algebraic expression.\n * Multiplication and division do not separate the terms of an algebraic expression.\n * Example: -7x^2y^3$is one term ($1 term).\n * Example: -7x^2 + y^3$has two terms ($2 terms).\n\n* **Types of Algebraic Expressions by Number of Terms**:\n * **Monomial**: An algebraic expression having only one term.\n * Example: -7x^2y^3$(Number of terms:$1$, Terms:$-7x^2y^3).\n * **Binomial**: An algebraic expression having two terms.\n * Example: 5x^2y - 7x$(Number of terms:$2$, Terms:$5x^2y$,$-7x).\n * **Trinomial**: An algebraic expression having three terms.\n * Example: -3xy^3 + 5xz^2 + \frac{11}{12}$(Number of terms:$3$, Terms:$-3xy^3$,$5xz^2$,$\frac{11}{12}).\n * **Multinomial**: An algebraic expression having two or more terms.\n * Example: 9x^5 - 3x^2 + 4 - \frac{11}{x}$(Number of terms:$4$, Terms:$9x^5$,$-3x^2$,$4$,$-\frac{11}{x}).\n\n* **Factors**:\n * Each quantity (constant or literal) multiplied together to form a product is called a factor of the product.\n * **Numerical Factor**: A factor consisting of a constant.\n * **Literal Factor**: A factor containing only literals.\n * Example: In -7xy^2:\n * The numerical factor is -7.\n * The literal factors are x$,$y$,$y^2$,$xy$, and$xy^2.\n\n* **Constant Term**:\n * The term of an algebraic expression having no literal factors is called its constant term.\n * Example: In the expression -3x^2y^3 + \frac{5}{x} - 7$, the constant term is$-7.\n * Example: The expression 9x^5 - 3x^2 + \frac{11}{x} has no constant term.\n\n# Coefficients, Like Terms, and Polynomials\n\n* **Coefficients**:\n * Any factor of a (non-constant) term of an algebraic expression is called the coefficient of the remaining factor of the term.\n * **Numerical Coefficient (or simply Coefficient)**: The constant part of a term.\n * **Literal Coefficient**: The remaining non-constant part of the term.\n * Example: Consider the expression 7p^3q^2 - 5p^2q - 3p + 2:\n * For the term -5p^2q:\n * The numerical coefficient = -5\n * The literal coefficient = p^2q\n * The coefficient of p^2$=$-5q\n * The coefficient of 5p$=$-pq\n * The coefficient of -q$=$5p^2\n * Implicit Coefficient Rule: When writing x$, it means$1x$. If no coefficient is written before a literal, the coefficient is always taken as$1.\n\n* **Like and Unlike Terms**:\n * **Like Terms**: Terms having the same literal coefficients.\n * Example: 5x^2yz$,$-3x^2yz$, and$\frac{3}{22}yzx^2 are like terms.\n * **Unlike Terms**: Terms having different literal coefficients.\n * Example: 7ab$,$-3a^2b$, and$ab^2 are unlike terms.\n\n* **Polynomials**:\n * An algebraic expression is called a polynomial if the powers of the variables involved in it in each term are non-negative integers.\n * A polynomial may contain any number of terms, one or more than one.\n * **Degree of a Polynomial**: Take the sum of the powers of the variables in each term; the greatest sum is the degree of the polynomial.\n * Examples:\n * 3x + 5$: Polynomial of two terms (binomial); degree$1\n * 8x^2y - 7xy^2 + 9xy + 3$: Polynomial of four terms (multinomial); degree$3\n * 5x^2 + 4x + 7$: Polynomial of three terms (trinomial); degree$2\n * 3x + 5y + xy + 5x^2y + 9$: Polynomial of five terms (multinomial); degree$3\n * 9x^5 + \frac{3}{x} + 4$: Not a polynomial because in the term$\frac{3}{x} = 3x^{-1}$, the power of$x$is a negative integer ($-1). However, it is a trinomial.\n\n* **Difference Between Multinomial and Polynomial**:\n * **Multinomial**:\n * Refers to the number of terms of an algebraic expression (an algebraic expression having two or more terms).\n * The powers of the variables involved in each term may be integers (negative, zero, or positive) or fractions.\n * **Polynomial**:\n * Refers to non-negative powers of the variables used in the algebraic expression.\n * May contain any number of terms, one or more than one term.\n * The powers of the variables involved in each term must be non-negative integers.\n\n# Addition of Algebraic Expressions\n\n* **Methods for Addition**:\n * **Horizontal Method**: Collect different groups of like terms and then find the sum of like terms in each group.\n * **Column Method**: Write each expression to be added in a separate row, placing like terms one below the other, and then add them.\n\n* **Examples of Addition**:\n * **Example 1(i)**: Add 3x^2 - 5xy + 4y^2 - 1$,$7y^2 - 9xy + 5$, and$7xy - 4x^2 + y^2 - 13\n * **Horizontal Method**:\n * Express sum: (3x^2 - 5xy + 4y^2 - 1) + (7y^2 - 9xy + 5) + (7xy - 4x^2 + y^2 - 13)\n * Group like terms: 3x^2 - 4x^2 - 5xy - 9xy + 7xy + 4y^2 + 7y^2 + y^2 - 1 + 5 - 13\n * Result: -x^2 - 7xy + 12y^2 - 9\n * **Column Method**:\n * Row 1: 3x^2 - 5xy + 4y^2 - 1\n * Row 2: - 9xy + 7y^2 + 5\n * Row 3: -4x^2 + 7xy + y^2 - 13\n * Sum: -x^2 - 7xy + 12y^2 - 9\n\n * **Example 1(ii)**: Add 7xy + 5yz - 3zx$,$4yz + 9zx - 4y$, and$-3xz + 5x - 2xy\n * Note: xz$is the same as$zx.\n * **Horizontal Method**:\n * Express sum: (7xy + 5yz - 3zx) + (4yz + 9zx - 4y) + (-3xz + 5x - 2xy)\n * Group like terms: 7xy - 2xy + 5yz + 4yz - 3zx + 9zx - 3zx + 5x - 4y\n * Result: 5xy + 9yz + 3zx + 5x - 4y\n * **Column Method**:\n * Row 1: 7xy + 5yz - 3zx\n * Row 2: + 4yz + 9zx - 4y\n * Row 3: -2xy - 3zx + 5x\n * Sum: 5xy + 9yz + 3zx + 5x - 4y\n\n# Subtraction of Algebraic Expressions\n\n* **Rule for Subtraction**:\n * To subtract algebraic expressions, change the sign of each term of the algebraic expression to be subtracted and then add. Use either horizontal method or column method.\n\n* **Examples of Subtraction**:\n * **Example 2(i)**: Subtract 5x^3 - 3x^2 - 8$from$2x^3 - 5x^2 - 11x + 2\n * **Horizontal Method**:\n * Express subtraction: (2x^3 - 5x^2 - 11x + 2) - (5x^3 - 3x^2 - 8)\n * Change signs of subtrahend: 2x^3 - 5x^2 - 11x + 2 - 5x^3 + 3x^2 + 8\n * Group like terms: 2x^3 - 5x^3 - 5x^2 + 3x^2 - 11x + 2 + 8\n * Result: -3x^3 - 2x^2 - 11x + 10\n * **Column Method**:\n * Row 1 (Minuend): 2x^3 - 5x^2 - 11x + 2\n * Row 2 (Subtrahend with changed signs): -5x^3 + 3x^2 + 8\n * Result: -3x^3 - 2x^2 - 11x + 10\n\n * **Example 2(ii)**: Subtract 5x^2 - 4y^2 + 6y - 3$from$7x^2 - 4xy + 8y^2 + 5x - 3y\n * **Horizontal Method**:\n * Express subtraction: (7x^2 - 4xy + 8y^2 + 5x - 3y) - (5x^2 - 4y^2 + 6y - 3)\n * Change signs of subtrahend: 7x^2 - 4xy + 8y^2 + 5x - 3y - 5x^2 + 4y^2 - 6y + 3\n * Group like terms: 7x^2 - 5x^2 - 4xy + 8y^2 + 4y^2 + 5x - 3y - 6y + 3\n * Result: 2x^2 - 4xy + 12y^2 + 5x - 9y + 3\n * **Column Method**:\n * Row 1 (Minuend): 7x^2 - 4xy + 8y^2 + 5x - 3y\n * Row 2 (Subtrahend with changed signs): -5x^2 + 4y^2 - 6y + 3\n * Result: 2x^2 - 4xy + 12y^2 + 5x - 9y + 3$$