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Door 4 · Grade 8

🧩 Algebraic Expressions

A term is a single part of an expression separated by \(+\) or \(-\) signs

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Terms, Factors & Types of Expressions

What are Algebraic Expressions?

An algebraic expression is a combination of constants, variables, and mathematical operations (addition, subtraction, multiplication, division, exponents). For example: \(3x + 5\), \(2a² - 4ab + b²\), \(7y - 3\)

What are Terms, Factors, and Coefficients?

• Term: A single number, variable, or product of numbers and variables separated by \(+\) or \(-\) signs
• Example: In \(4x² + 3xy - 7\), the terms are \(4x²\), \(3xy\), and \(-7\)

• Factor: Numbers or variables multiplied together to form a term
• Example: In \(6xy\), factors are \(6\), \(x\), and \(y\)

• Coefficient: The numerical factor of a term (the number multiplied by the variable)
• Example: In \(5x²\), the coefficient is \(5\)

Types of Algebraic Expressions:

TypeDefinitionNumber of TermsExamplesMonomialExpression with exactly one term1\(7x\), \(-3y²\), \(5ab\),…

Worked example

Problem: Example 1: Identify the terms, coefficients, and constant term in the expression \(7x² - 3xy + 5y - 9\)

  1. - Step 1: Terms are: \(7x²\), \(-3xy\), \(5y\), and \(-9\)
  2. - Step 2: Coefficients: For \(7x²\) → \(7\); For \(-3xy\) → \(-3\); For \(5y\) → \(5\)
  3. - Step 3: Constant term (term without variable): \(-9\)

Answer: - Step 3: Constant term (term without variable): \(-9\)

Adding, Subtracting & Multiplying Expressions

Addition and Subtraction of Algebraic Expressions

To add or subtract algebraic expressions:

• Group like terms together
• Add or subtract the coefficients of like terms
• Keep unlike terms as they are

Example: \((3x² + 2x - 5) + (4x² - 3x + 7) = (3x² + 4x²) + (2x - 3x) + (-5 + 7) = 7x² - x + 2\)

Multiplication of Algebraic Expressions

TypeRuleExampleMonomial × MonomialMultiply coefficients, add exponents of like variables\((3x²)(4x³) = 12x^5\)Monomial × PolynomialDistributive property: Multiply monomial by each term\(2x(x + 3) = 2x² + 6x\)Binomial × BinomialFOIL method or distributive property\((x + 2)(x + 3) = x² + 5x + 6\)Polynomial × PolynomialMultiply each term of first by each term of second\((x + 1)(x² + 2x + 3) = x³ + 3x² + 5x + 3\)
Division of Algebraic Expressions

• Monomial ÷ Monomial: Divide coefficients, subtract exponents of like variables
•…

Worked example

Problem: Example 1: Add: \((5a² - 3ab + 2b²)\) and \((-2a² + 7ab - b²)\)

  1. - Step 1: Group like terms together
  2. - Step 2: \(a²\) terms: \(5a² + (-2a²) = 3a²\)
  3. - Step 3: \(ab\) terms: \((-3ab) + 7ab = 4ab\)
  4. - Step 4: \(b²\) terms: \(2b² + (-b²) = b²\)
  5. - Step 5: Combine all: \(3a² + 4ab + b²\)

Answer: - Step 5: Combine all: \(3a² + 4ab + b²\)

Standard Identities & Applications

What are Algebraic Identities?

Algebraic identities are equations that are true for all values of the variables. They help us simplify complex algebraic expressions and perform calculations mentally.

The Three Standard Identities:

IdentityFormulaNameIdentity I\((a + b)² = a² + 2ab + b²\)Square of a SumIdentity II\((a - b)² = a² - 2ab + b²\)Square of a DifferenceIdentity III\((a + b)(a - b) = a² - b²\)Difference of Squares
Verification of Identities:

For Identity I: \((a + b)² = (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b²\)

Applications of Identities:

• Expanding expressions: \((3x + 5)² = (3x)² + 2(3x)(5) + 5² = 9x² + 30x + 25\)
• Factorizing expressions: \(x² - 9 = (x)² - (3)² = (x + 3)(x - 3)\)
• Simplifying calculations mentally: \(102² = (100 + 2)² = 100² + 2(100)(2) + 2² = 10000 + 400 + 4 = 10404\)

Mental Mathematics Using Identities:

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Worked example

Problem: Example 1: Using identity, find the value of \((5x + 3y)²\)

  1. - Step 1: Identify \(a = 5x\) and \(b = 3y\)
  2. - Step 2: Use identity \((a + b)² = a² + 2ab + b²\)
  3. - Step 3: \(a² = (5x)² = 25x²\)
  4. - Step 4: \(2ab = 2 × 5x × 3y = 30xy\)
  5. - Step 5: \(b² = (3y)² = 9y²\)

Answer: - Step 5: \(b² = (3y)² = 9y²\)