6.2.3 Algebraic Expressions (III), PT3 Practice

Posted on February 23, 2018 by user

Question 11:
(a)
Expand: x (2 + y)

(b)

$\begin{array}{l} Express \frac{5}{6 y} - \frac{3 x - 5}{12 y} as a single \\ fraction in its simplest form . \end{array}$

Solution:
(a)
x (2 + y) = 2x + xy

(b)

$\begin{array}{l} \frac{5}{6 y} - \frac{3 x - 5}{12 y} \\ = \frac{5 \times 2}{6 y \times 2} - \frac{3 x - 5}{12 y} \\ = \frac{10 - (3 x - 5)}{12 y} \\ = \frac{10 - 3 x + 5}{12 y} \\ = \frac{15 - 3 x}{12 y} \\ = \frac{3 (5 - x)}{12^{4} y} \\ = \frac{5 - x}{4 y} \end{array}$

6.2.2 Algebraic Expressions (III), PT3 Practice

Posted on February 22, 2018 by user

6.2.2 Algebraic Expressions (III), PT3 Practice

Question 6:
(a) Simplify each of the following:

$\begin{array}{l} (i) \frac{12 m n}{32} \\ (ii) \frac{x^{2} - x y}{x} \end{array}$
(b) Express

$\frac{1}{2 q} - \frac{2 p - 7}{6 q}$ as a single fraction in its simplest form.

Solution:

$\begin{array}{l} (a)(i) \frac{12 m n}{32} = \frac{3 m n}{8} \\ (a)(ii) \frac{x^{2} - x y}{x} = \frac{x (x - y)}{x} = x - y \end{array}$

(b)

$\begin{array}{l} \frac{1}{2 q} - \frac{2 p - 7}{6 q} = \frac{1 \times 3}{2 q \times 3} - \frac{(2 p - 7)}{6 q} \\ = \frac{3 - 2 p + 7}{6 q} \\ = \frac{10 - 2 p}{6 q} \\ = \frac{2 (5 - p)}{36 q} \\ = \frac{5 - p}{3 q} \end{array}$

Question 7:

(a) Factorise 2ae + 3af – 6de – 9df

(b) Simplify

$\frac{a^{2} - b^{2}}{{(a + b)}^{2}}$

Solution:

(a)

2ae + 3af – 6de – 9df = a (2e + 3f ) – 3d (2e + 3f)

= (2e + 3f ) (a – 3d)

(b)

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

Question 8:

(a) Factorise –8c² – 12ac.

(b) Simplify

$\frac{a e + a d - 2 b e - 2 b d}{a^{2} - 4 b^{2}} .$

Solution:

(a)

–8c²– 12ac

= –4c (2c + 3a)

(b)

$\begin{array}{l} \frac{a e + a d - 2 b e - 2 b d}{a^{2} - 4 b^{2}} = \frac{a (e + d) - 2 b (e + d)}{(a + 2 b) (a - 2 b)} \\ = \frac{(e + d) (a - 2 b)}{(a + 2 b) (a - 2 b)} \\ = \frac{e + d}{a + 2 b} \end{array}$

Question 9:

(a) Factorise 12x² – 27y²

(b) Simplify

$\frac{3 m^{2} - 10 m + 3}{m^{2} - 9} \div \frac{3 m - 1}{m + 3} .$

Solution:

(a)

12x²– 27y²= 3 (4x² – 9y²)

= 3(2x – 3y) (2x + 3y)

(b)

$\begin{array}{l} \frac{3 m^{2} - 10 m + 3}{m^{2} - 9} \div \frac{3 m - 1}{m + 3} = \frac{(3 m - 1) (m - 3)}{(m + 3) (m - 3)} \times \frac{m + 3}{3 m - 1} \\ = 1 \end{array}$

Question 10:

$Simplify: \frac{8 m + m n}{3 m} \div \frac{n^{2} - 64}{24}$

Solution:

$\begin{array}{l} \frac{8 m + m n}{3 m} \div \frac{n^{2} - 64}{24} \\ = \frac{8 m + m n}{3 m} \times \frac{24}{n^{2} - 64} \\ = \frac{m (8 + n)}{3 m} \times \frac{24}{n^{2} - 8^{2}} \\ = \frac{m (8 + n)}{3 m} \times \frac{24_{8}}{(n - 8) (n + 8)} \\ = \frac{8}{n - 8} \end{array}$

6.2.1 Algebraic Expressions (III), PT3 Practice

Posted on February 20, 2018 by user

6.2.1 Algebraic Expressions (III), PT3 Practice

Question 1:

(a)(i) Factorise 18a + 3

(a)(ii) Expand –3 (–y + 5)

(b) Express

$\frac{5}{6 y} - \frac{3 x - 5}{12 y}$ as a single fraction in its simplest form.

Solution:

(a)(i) 18a + 3 = 3(6a + 1)

(a)(ii) –3 (–y + 5) = 3y – 15

(b)

$\begin{array}{l} \frac{5}{6 y} - \frac{3 x - 5}{12 y} = \frac{5 \times 2}{6 y \times 2} - \frac{(3 x - 5)}{12 y} \\ = \frac{10 - 3 x + 5}{12 y} \\ = \frac{15 - 3 x}{12 y} \\ = \frac{3 (5 - x)}{412 y} \\ = \frac{5 - x}{4 y} \end{array}$

Question 2:

(a) Expand:

(i) 3 (–a + c)

(ii) –5 (a – c)

(b) Factorise 4x + 2

$\frac{3 x + 6}{x^{2} - 4} \div \frac{x + 2}{x - 2}$

Solution:

(a)(i) 3 (–a + c) = –3a + 3c

(a)(ii) –5 (a – c) = –5a + 5c

(c)

$\begin{array}{l} \frac{3 x + 6}{x^{2} - 4} \div \frac{x + 2}{x - 2} = \frac{3 (x + 2)}{(x + 2) (x - 2)} \times \frac{x - 2}{x + 2} \\ = \frac{3}{x + 2} \end{array}$

Question 3:

(a) Factorise:

(i) 5m + 25

(ii) 7x + 9xy

(b) Simplify:

$\frac{4 x - 12}{4 y} \div \frac{x^{2} - 9}{y z}$

Solution:

(a)(i)5m + 25 = 5 (m + 5)

(a)(ii)7x + 9xy = x (7 + 9y)

$\begin{array}{l} (b) \frac{4 x - 12}{4 y} \div \frac{x^{2} - 9}{y z} = \frac{4 (x - 3)}{4 y} \times \frac{y z}{(x + 3) (x - 3)} \\ = \frac{z}{x + 3} \end{array}$

Question 4:

(a) Factorise completely:

4 – 100n²

(b) Express

$\frac{4}{5 x} - \frac{7 - 10 y}{15 x}$ as a single fraction in its simplest form.

Solution:

(a)

4 – 100n²= (2 + 10n)(2 – 10n)

(b)

$\begin{array}{l} \frac{4}{5 x} - \frac{7 - 10 y}{15 x} = \frac{4 \times 3}{5 x \times 3} - \frac{(7 - 10 y)}{15 x} \\ = \frac{12 - 7 + 10 y}{15 x} \\ = \frac{5 + 10 y}{15 x} \\ = \frac{5 (1 + 2 y)}{315 x} \\ = \frac{1 + 2 y}{3 x} \end{array}$

Question 5:

(a) Simplify:

(m – 4n)(m + 4n) – m²

(b) Simplify:

$\frac{3 x - 3 y}{x + y} \times \frac{2 x + 2 y}{6 x}$

Solution:

(a)

(m – 4n)(m + 4n) – m²

= m²+ 4mn – 4mn – 4n² – m²

= 0

(b)

$\begin{array}{l} \frac{3 x - 3 y}{x + y} \times \frac{2 x + 2 y}{6 x} = \frac{3 (x - y)}{x + y} \times \frac{2 (x + y)}{6 x} \\ = \frac{x - y}{x} \end{array}$

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:

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