Algebraic Expressions

Grade 12 Math - Algebraic Expressions

Lesson Objectives

Expand algebraic expressions using distributive and binomial rules.
Factorize quadratic and complex algebraic expressions.
Simplify complex algebraic fractions using common factors and identities.

Lesson Introduction

In this lesson, we will learn how to expand and factorize algebraic expressions, as well as simplify algebraic fractions. Mastery of these skills is essential for solving more advanced algebraic problems in mathematics and related subjects.

Core Lesson Content

Worked Example

Expanding Expressions

Use the distributive law and identities such as:

$(a + b)^2 = a^2 + 2ab + b^2$
$(a - b)^2 = a^2 - 2ab + b^2$
$(a + b)(a - b) = a^2 - b^2$

Example 1: Expand

(2x + 3)^2

(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9

Example 2: Expand

(x - 5)(x + 7)

(x - 5)(x + 7) = x^2 + 7x - 5x - 35 = x^2 + 2x - 35

Factorizing Expressions

Types of factorization:

Common factor
Difference of squares
Trinomials
Grouping

Example 3: Factorize

4x^2 - 25

4x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5)

Example 4: Factorize

x^2 + 5x + 6

Find two numbers that multiply to 6 and add up to 5: 2 and 3

x^2 + 5x + 6 = (x + 2)(x + 3)

Example 5: Factorize

3x^2 + 11x + 6

Split the middle term:

3x^2 + 2x + 9x + 6 = (3x^2 + 2x) + (9x + 6)

Factor by grouping:

x(3x + 2) + 3(3x + 2) = (x + 3)(3x + 2)

Simplifying Algebraic Fractions

Steps:

Factor numerator and denominator if possible.
Cancel common factors.

Example 6: Simplify

\frac{x^2 - 9}{x^2 - x - 6}

Factor numerator:

x^2 - 9 = (x - 3)(x + 3)

Factor denominator:

x^2 - x - 6 = (x - 3)(x + 2)

Simplify:

\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} = \frac{x + 3}{x + 2}

Example 7: Simplify

\frac{4x^2 - 1}{2x - 1}

4x^2 - 1 = (2x - 1)(2x + 1)

Simplify:

\frac{(2x - 1)(2x + 1)}{2x - 1} = 2x + 1

Exercises

Expand and simplify $(3x - 2)^2$
[WAEC] Expand $(x + 4)(x - 3)$
Factorize $x^2 - 10x + 21$
[NECO] Factorize $2x^2 + 7x + 3$
Factorize completely: $x^3 - 3x^2 - x + 3$
Simplify $\frac{x^2 - 4}{x^2 + 2x}$
Simplify $\frac{2x^2 + 5x + 2}{x + 2}$
Factor and simplify: $\frac{3x^2 - 12}{x^2 - 4}$
[WAEC] Simplify the expression: $\frac{x^2 - 16}{x^2 + 3x - 4}$
Simplify $\frac{x^2 - 6x + 9}{x^2 - 9}$

Conclusion/Recap

Being able to expand, factorize, and simplify algebraic expressions and fractions is crucial for solving complex equations and real-world problems. Always look for opportunities to factor and cancel to simplify your expressions fully.