6.1 Algebraic Expressions III

Posted on February 15, 2018 by user

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= a² + 2ab + b²
(a – b)²= a² – 2ab + b²
(a + b) (a – b) = a² – b²

6.1.2 Factorization

1. Factorize algebraic expressions:

ab + ac = a(b + c)
a²– b²= (a + b) (a – b)
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:

$\begin{array}{l} • \frac{^{1} 4 b c}{^{_{^{3}}} 12 b d} = \frac{c}{3 d} \\ • \frac{b m + b n}{e m + e n} = \frac{b (m + n)}{e (m + n)} \\ = \frac{b}{e} \end{array}$

3(b) Simplification of algebraic fractions by using difference of two squares:

$\begin{array}{l} \frac{a^{2} - b^{2}}{a n + b n} = \frac{(a + b) (a - b)}{n (a + b)} \\ = \frac{a - b}{n} \end{array}$

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:

$\frac{a}{m} + \frac{b}{n} = \frac{a n + b m}{n m}$

6.1.4 Multiplication and Division of Algebraic Fractions

1. Without simplification:

$\begin{array}{l} • \frac{a}{m} \times \frac{b}{n} = \frac{a b}{m n} \\ • \frac{a}{m} \div \frac{b}{n} = \frac{a}{m} \times \frac{n}{b} \\ = \frac{a n}{b m} \end{array}$

2. With simplification:

$\begin{array}{l} • \frac{a}{c m} \times \frac{b m}{d} = \frac{a b}{c d} \\ • \frac{a}{c m} \div \frac{b}{d m} = \frac{a}{c m} \times \frac{d m}{b} \\ = \frac{a d}{b c} \end{array}$