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Algebra - Like and Unlike Terms, Monomials, Binomials, Trinomials

Grade 7ICSE

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

🔑Concepts

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Algebraic Expression and Terms: An algebraic expression is a combination of variables and constants connected by arithmetic operations. Terms are the individual parts separated by ++ or −- signs. Imagine a horizontal chain where each link represents a term; for example, in the expression 5x2−3y+75x^2 - 3y + 7, the terms are 5x25x^2, −3y-3y, and 77.

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Monomial: An expression that consists of only one non-zero term. Visually, this is a single block or cluster of numbers and variables multiplied together without any addition or subtraction signs between them. Examples include 4ax4ax, −7-7, or 23y2\frac{2}{3}y^2.

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Binomial: An expression containing exactly two terms. Visualize this as two distinct blocks connected by a single plus or minus sign. For example, in 3x+5y3x + 5y, the two terms 3x3x and 5y5y are the two parts of the binomial.

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Trinomial: An expression containing exactly three terms. Think of it as a trio of components linked together, such as a2+2ab+b2a^2 + 2ab + b^2. Each term is separated by a mathematical operator, forming a three-part structure.

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Polynomial: A general term for an algebraic expression with one or more terms. While monomials, binomials, and trinomials are specific names, any expression like x3−4x2+x−9x^3 - 4x^2 + x - 9 with multiple terms and non-negative integer exponents is a polynomial.

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Like Terms: Terms that have the same literal coefficients, meaning they have the identical variables raised to the same powers. Imagine like terms as identical fruits in a basket; for instance, 5x2y5x^2y and −2x2y-2x^2y are like terms because their variable parts x2yx^2y are a perfect match.

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Unlike Terms: Terms that have different variables or the same variables raised to different powers. Visualize these as different types of objects, like an apple (2a2a) and an orange (3b3b), which cannot be merged into a single count of one item. For example, 4x4x and 4x24x^2 are unlike terms because the exponents of xx differ.

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Factors and Coefficients: In a term like −5xy-5xy, the numerical part −5-5 is the numerical coefficient, while xx and yy are literal factors. The coefficient of a term is the product of all factors except the one being considered.

📐Formulae

General form of a Monomial=axn\text{General form of a Monomial} = ax^n

General form of a Binomial=axn+bxm\text{General form of a Binomial} = ax^n + bx^m

General form of a Trinomial=axn+bxm+cxp\text{General form of a Trinomial} = ax^n + bx^m + cx^p

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

💡Examples

Problem 1:

Identify the like terms in the following expression: 7x2y−3xy2+5x2y+10x−2x7x^2y - 3xy^2 + 5x^2y + 10x - 2x.

Solution:

Step 1: Group terms with identical variables and powers.

  • The term 7x2y7x^2y has variables x2x^2 and yy.
  • The term 5x2y5x^2y also has variables x2x^2 and yy.
  • The term 10x10x has the variable xx.
  • The term −2x-2x also has the variable xx. Step 2: List the pairs of like terms.
  • Pair 1: 7x2y7x^2y and 5x2y5x^2y
  • Pair 2: 10x10x and −2x-2x

Explanation:

Like terms are identified by matching their literal (variable) parts exactly, regardless of their numerical coefficients.

Problem 2:

Classify the expression 3a2−5b+83a^2 - 5b + 8 and identify its terms and the numerical coefficient of the second term.

Solution:

Step 1: Count the number of terms. The terms are 3a23a^2, −5b-5b, and 88. Since there are 3 terms, it is a Trinomial. Step 2: Identify the second term, which is −5b-5b. Step 3: Extract the numerical part of the second term. The numerical coefficient is −5-5.

Explanation:

An expression is classified based on the count of terms separated by plus or minus signs. The numerical coefficient includes the sign attached to the term.