# 6.1 Algebraic Expressions III

6.1 Algebraic Expressions III

6.1.1 Expansion
1. The product of an algebraic term and an algebraic expression:
• a(b + c) = ab + ac
•  a(bc) = ab ac
2. The product of an algebraic expression and another algebraic expression:
• (a + b) (c + d)  = ac + ad + bc + bd
• (a + b)2= a2 + 2ab + b2
• (ab)2= a2 – 2ab + b2
• (a + b) (ab) = a2b2

6.1.2 Factorization
1. Factorize algebraic expressions:
•  ab + ac = a(b + c)
• a2b2 = (a + b) (ab)
• a2+ 2ab + b2 = (a + b)2
• ac + ad + bc + bd = (a + b) (c + d)
2. Algebraic fractions are fractions where both the numerator and the denominator or either the numerator or the denominator are algebraic terms or algebraic expressions.
Example:
$\frac{3}{b},\frac{a}{7},\frac{a+b}{a},\frac{b}{a-b},\frac{a-b}{c+d}$

3(a) Simplification of algebraic fractions by using common factors:
$\begin{array}{l}•\text{}\frac{{}^{1}\overline{)4}\overline{)b}c}{{}^{{}_{{}^{3}}}\overline{)12}\overline{)b}d}=\frac{c}{3d}\\ •\text{}\frac{bm+bn}{em+en}=\frac{b\overline{)\left(m+n\right)}}{e\overline{)\left(m+n\right)}}\\ \text{}=\frac{b}{e}\end{array}$

3(b) Simplification of algebraic fractions by using difference of two squares:
$\begin{array}{l}\frac{{a}^{2}-{b}^{2}}{an+bn}=\frac{\overline{)\left(a+b\right)}\left(a-b\right)}{n\overline{)\left(a+b\right)}}\\ \text{}=\frac{a-b}{n}\end{array}$

6.1.3 Addition and Subtraction of Algebraic Fractions
1. If they have a common denominator:
$\frac{a}{m}+\frac{b}{m}=\frac{a+b}{m}$

2.
If they do not have a common denominator:
$\frac{a}{m}+\frac{b}{n}=\frac{an+bm}{nm}$

6.1.4 Multiplication and Division of Algebraic Fractions
1. Without simplification:
$\begin{array}{l}•\text{}\frac{a}{m}×\frac{b}{n}=\frac{ab}{mn}\\ •\text{}\frac{a}{m}÷\frac{b}{n}=\frac{a}{m}×\frac{n}{b}\\ \text{}=\frac{an}{bm}\end{array}$

2.
With simplification:
$\begin{array}{l}•\text{}\frac{a}{c\overline{)m}}×\frac{b\overline{)m}}{d}=\frac{ab}{cd}\\ •\text{}\frac{a}{cm}÷\frac{b}{dm}=\frac{a}{c\overline{)m}}×\frac{d\overline{)m}}{b}\\ \text{}=\frac{ad}{bc}\end{array}$