Introduction to Algebra: Terms, Coefficients, Degree of a Term, and Degree of an Expression
Welcome to Unit 2 of Preparatory 1 Mathematics! In this unit, we transition from concrete numerical arithmetic to the powerful, abstract domain of Algebra. An 'algebraic term' is defined as the product of one or more constants and variables (e. g., 5x, -3x²y). It consists of two essential parts: the numerical constant known as the 'numerical coefficient', and the algebraic factors (variables). The 'degree of an algebraic term' is calculated by summing the exponents of all the variable factors composing that term: the term 5x is of the 1st degree, while -3x²y is of the 3rd degree (2 + 1 = 3). Any non-zero real number without variables (e. g., 7) is a 'constant term' of the 0th degree (since 7 = 7x⁰), whereas the number 0 has no defined degree. An 'algebraic expression' is formed by one or more algebraic terms separated by addition (+) or subtraction (-) signs. The 'degree of an algebraic expression' is strictly defined as the highest degree among all its simplified constituent terms.
1. Anatomy and Degree of an Algebraic Term
Core algebraic definitions and rules:
- Algebraic Term: Product of numbers and letters (e.g., 4ab, -7x³y²).
- Numerical Coefficient: The constant multiplier; in -7x³y², the coefficient is -7.
- Degree of a Term: Sum of the exponents of its variable factors:
• For -7x³y²: 3 + 2 = 5th degree.
• For 8x: exponent is 1 ← 1st degree.
• For 12 (constant): 0th degree (12x⁰).
2. Algebraic Expressions and Their Degrees
Classification and degree determination:
- Monomial: An expression with exactly one term (e.g., 3x²).
- Binomial: An expression with two terms (e.g., 2x + 5).
- Trinomial: An expression with three terms (e.g., x² - 4x + 7).
- Degree Rule: The degree of an expression is the maximum degree of its individual terms:
• In 5x²y - 3x⁴ + 7y: term degrees are 3, 4, 1. The highest is 4, so the expression is of the 4th degree. - Standard Order: Arranging terms in descending or ascending powers of a specified variable.
Step-by-Step Algebraic Workshop:
(a) Coefficient = -5; Degree = 2 + 3 = 5 (5th degree).
(b) Coefficient = 8; Degree = 1 (1st degree).
(c) Coefficient = -19; Degree = 0 (0th degree, constant term).
Descending order: 2x³ - 5x² + 4x - 7.
The highest exponent is 3; hence, the expression is of the 3rd degree.
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