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Algebra - Algebraic Expressions: Terms, Factors, Coefficients

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An algebraic expression is a combination of constants and variables connected by basic arithmetic operators like addition, subtraction, multiplication, and division. Imagine an expression as a mathematical sentence where variables like xx or yy represent unknown values and constants represent fixed values.

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Terms are the individual parts of an algebraic expression separated by ++ or −- signs. For instance, in the expression 4x2−3xy+74x^2 - 3xy + 7, there are three terms: 4x24x^2, −3xy-3xy, and 77. You can visualize this as a 'Tree Diagram' where the expression is the root, and each term is a primary branch extending from it.

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Factors are the components that are multiplied together to form a term. In the term 5xy5xy, the factors are 55, xx, and yy. If you break down a term using a factor tree, you would see the numerical constant and each literal variable as separate leaves at the end of the branches.

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A Coefficient is the numerical factor of a term. In the term −8ab-8ab, the numerical coefficient is −8-8. If a term appears as just x2x^2, its coefficient is understood to be 11, and for −y-y, the coefficient is −1-1. Visualize the coefficient as the 'weight' or 'multiplier' attached to the variable parts.

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A Constant Term is a term in an algebraic expression that does not contain any variables. Its value remains fixed regardless of what the variables represent. In 2x+52x + 5, the number 55 is the constant term. It stands alone like a fixed point on a number line.

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Like Terms are terms that have the exact same variable parts with the same exponents. For example, 3x2y3x^2y and −7x2y-7x^2y are like terms. Unlike Terms have different variables or different powers, such as 5x5x and 5x25x^2. Visualize like terms as objects of the same shape and size that can be grouped or 'merged' together through addition or subtraction.

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Expressions are classified by the number of terms they contain. A Monomial has one term (e.g., 5x25x^2), a Binomial has two terms (e.g., a+ba + b), and a Trinomial has three terms (e.g., x2+2x+1x^2 + 2x + 1). Any expression with one or more terms is generally called a Polynomial.

📐Formulae

Expression=Term1+Term2+⋯+Termn\text{Expression} = \text{Term}_1 + \text{Term}_2 + \dots + \text{Term}_n

Term=(Numerical Coefficient)×(Literal Variables)\text{Term} = (\text{Numerical Coefficient}) \times (\text{Literal Variables})

Addition of Like Terms:axn+bxn=(a+b)xn\text{Addition of Like Terms}: ax^n + bx^n = (a + b)x^n

Subtraction of Like Terms:axn−bxn=(a−b)xn\text{Subtraction of Like Terms}: ax^n - bx^n = (a - b)x^n

Product of Factors:2×x×x×y=2x2y\text{Product of Factors}: 2 \times x \times x \times y = 2x^2y

💡Examples

Problem 1:

Identify the terms and their factors in the expression: 12x2−5xy12x^2 - 5xy.

Solution:

  1. Identify the terms: The expression is composed of two parts separated by a minus sign. The terms are 12x212x^2 and −5xy-5xy. \n2. Break down the first term: For 12x212x^2, the factors are 1212, xx, and xx. \n3. Break down the second term: For −5xy-5xy, the factors are −5-5, xx, and yy.

Explanation:

Terms are identified by looking at the segments separated by addition or subtraction operators. Factors are the individual elements (numbers and variables) that multiply together to make that specific term.

Problem 2:

In the expression 7x2y−4x+97x^2y - 4x + 9, identify the numerical coefficient of each term and state the constant term.

Solution:

  1. First term is 7x2y7x^2y: The numerical coefficient is 77. \n2. Second term is −4x-4x: The numerical coefficient is −4-4. \n3. Third term is 99: This term has no variable, so it is the constant term.

Explanation:

The numerical coefficient is the number (including its sign) that multiplies the variables. The constant term is the part of the expression that contains only a number without any literal (variable) factors.