2.2.1 Factorisation and Algebraic Fractions, PT3 Practice - Mathematics

2.2.1 Factorisation and Algebraic Fractions, 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}

$\frac{5}{6 y} - \frac{3 x - 5}{12 y}$ 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{matrix} \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{matrix}

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

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}

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

Solution:

(a)(i) 3 (–a + c) = –3a + 3c

(a)(ii) –5 (a – c) = –5a + 5c

(c)

\begin{matrix} \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{matrix}

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

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}

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

Solution:

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

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

\begin{matrix} (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{matrix}

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

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)

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

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