2.2.3 Factorisation and Algebraic Fractions, PT3 Practice - Mathematics

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} .

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

Solution:

(a)

12x²– 27y²= 3 (4x² – 9y²)

= 3(2x – 3y) (2x + 3y)

(b)

\begin{matrix} \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{matrix}

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

Question 10:

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

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

Solution:

\begin{matrix} \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{matrix}

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

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

(b)

Express 56y−3x−512y as a single fraction in its simplest form.

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

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

(b)

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

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