Algebraic Expressions • Topic 2 of 3

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

Multiplication of Algebraic Expressions

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

Division of Algebraic Expressions

  • Monomial ÷ Monomial: Divide coefficients, subtract exponents of like variables
  • Example: \(\frac{15x^5}{3x^2} = 5x^{3}\)
  • Polynomial ÷ Monomial: Divide each term by the monomial
  • Example: \(\frac{6x^3 + 9x^2}{3x} = 2x^2 + 3x\)
  • Polynomial ÷ Binomial: Use long division method

Simplification of Algebraic Expressions

Simplification means combining like terms, applying distributive property, and reducing to the simplest form without changing the value.

Factorization Basics

Factorization is the process of writing an expression as a product of its factors (the reverse of multiplication).

Methods of Factorization:

  • Taking out common factors: \(4x^2 + 6x = 2x(2x + 3)\)
  • Grouping: \(ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)\)
  • Using identities: \(x^2 + 6x + 9 = (x + 3)^2\)
Adding & Subtracting Algebraic ExpressionsStep 1: Write expressions(3x + 2y − 4) + (2x − y + 6)Step 2: Group like terms(3x + 2x) + (2y − y) + (−4 + 6)Step 3: Simplify5x + y + 2Multiplication: Use FOIL or distributive law — (a+b)(c+d) = ac+ad+bc+bdDivision: Divide each term by the divisor separately
Solved Examples
1
Worked Example
Example 1: Add: \((5a^2 - 3ab + 2b^2)\) and \((-2a^2 + 7ab - b^2)\)
Solution- Step 1: Group like terms together - Step 2: \(a^2\) terms: \(5a^2 + (-2a^2) = 3a^2\) - Step 3: \(ab\) terms: \((-3ab) + 7ab = 4ab\) - Step 4: \(b^2\) terms: \(2b^2 + (-b^2) = b^2\) - Step 5: Combine all: \(3a^2 + 4ab + b^2\)
2
Worked Example
Example 2: Multiply: \((2x - 3)(x^2 + 4x - 5)\)
Solution- Step 1: Multiply each term of first expression with each term of second - Step 2: \(2x \times x^2 = 2x^3\) - Step 3: \(2x \times 4x = 8x^2\) - Step 4: \(2x \times (-5) = -10x\) - Step 5: \((-3) \times x^2 = -3x^2\) - Step 6: \((-3) \times 4x = -12x\) - Step 7: \((-3) \times (-5) = 15\) - Step 8: Combine like terms: \(2x^3 + (8x^2 - 3x^2) = 2x^3 + 5x^2\) - Step 9: \((-10x - 12x) = -22x\), then \(+15\) - Step 10: Final expression: \(2x^3 + 5x^2 - 22x + 15\)
3
Worked Example
Example 3: Factorize completely: \(4x^2y + 8xy^2 + 12xy\)
Solution- Step 1: Find the greatest common factor (GCF) of all terms - Step 2: Numbers: \(4, 8, 12\) have GCF = \(4\) - Step 3: Variables: \(x^2y\), \(xy^2\), \(xy\) have GCF = \(xy\) - Step 4: Overall GCF = \(4xy\) - Step 5: Divide each term by \(4xy\): - \(4x^2y ÷ 4xy = x\) - \(8xy^2 ÷ 4xy = 2y\) - \(12xy ÷ 4xy = 3\) - Step 6: Write as product: \(4xy(x + 2y + 3)\)

Key Points

  • Addition/Subtraction: Only like terms can be added or subtracted
  • Multiplication: Multiply coefficients, add exponents of same variables
  • FOIL method: First, Outer, Inner, Last for multiplying two binomials
  • Division: Subtract exponents for monomials; use long division for polynomials
  • Simplification: Combine like terms and remove brackets using distributive property
  • Factorization: Reverse of multiplication - express as product of factors
  • Always check if the result can be simplified further
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.$3x+5x=$
Explanation: Add like terms.
Q2.$7y-2y=$
Explanation: $5y$.
Q3.$2x\times4x=$
Explanation: $8x^2$.
Q4.$x(x+3)=$
Explanation: Distribute.
Practice more MCQs → 📝 Assignment · 35 marks