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(b – c) = ab – ac
2. The product of an algebraic expression and another algebraic expression:
- (a + b) (c + d) = ac + ad + bc + bd
- (a + b)^{2}= a^{2} + 2ab + b^{2}
- (a – b)^{2}= a^{2} – 2ab + b^{2}
- (a + b) (a – b) = a^{2} – b^{2}
6.1.2 Factorization
1. Factorize algebraic expressions:
- ab + ac = a(b + c)
- a^{2}– b^{2 }= (a + b) (a – b)
- a^{2}+ 2ab + b^{2 }= (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:
$\begin{array}{l}\u2022\text{}\frac{{}^{1}\overline{)4}\overline{)b}c}{{}^{{}_{{}^{3}}}\overline{)12}\overline{)b}d}=\frac{c}{3d}\\ \u2022\text{}\frac{bm+bn}{em+en}=\frac{b\overline{)\left(m+n\right)}}{e\overline{)\left(m+n\right)}}\\ \text{}=\frac{b}{e}\end{array}$
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:
6.1.4 Multiplication and Division of Algebraic Fractions
1. Without simplification:
$\begin{array}{l}\u2022\text{}\frac{a}{m}\times \frac{b}{n}=\frac{ab}{mn}\\ \u2022\text{}\frac{a}{m}\xf7\frac{b}{n}=\frac{a}{m}\times \frac{n}{b}\\ \text{}=\frac{an}{bm}\end{array}$
2. With simplification: