Topic 3 Factorising

3.1 Multiplying out brackets

\[\begin{align*} 3\cdot(2+4) &= 3\cdot 6 = 18\\ =\quad\quad\\ 3\cdot2 + 3\cdot 4 &= 6+12 = 18 \end{align*}\]

Figure 3.1: Illustrate distributivity with a rectangle area

General rules: \(a(b+c)=ab+ac\) and \(a(b-c) = ab-ac\).

Example 3.1 \(4(6x-2)=4\cdot 6x - 4\cdot 2 = 24x-12\).

Example 3.2 \((x+2y)(y-1)=(x+2y)\cdot y - (x+2y)\cdot 1\) \(= xy+2y\cdot y - x -2y\) \(= xy-2y^2-x-2y\).

Example 3.3 \((a-b+3)(4a+2)=(a-b+3)\cdot4a + (a-b+3)\cdot 2\) \(=a\cdot 4a - b\cdot 4a + 3\cdot 4a + a\cdot 2 - b\cdot 2 + 3\cdot 2\) \(= 4a^2 - 4ab + 14a - 2b+6\).

Example 3.4 \((3x+1)(y-2)(x+y)=(3x+1)\cdot(y-2)\cdot x + (3x+1)\cdot(y-2)\cdot y\) \(=(3x+1)\cdot(yx-2x) + (3x+1)\cdot(y^2-2y)\) \(= (3x+1)\cdot yx - (3x+1)\cdot 2x + (3x+1)\cdot y^2 - (3x+1)\cdot 2y\) \(= 3x^2y + xy - 6x^2 - 2x + 3xy^2 + y^2 - 6xy - 2y\).

3.2 Factorising

Factors are the individual constituents of a product expression.

For example: \(1\cdot 2\cdot 2\cdot 3\),    \(x(y+z)\),    \((a+b)(b+3)\)

Factorising is writing something as a product.

For example: \(12 = 1\cdot 2\cdot 2\cdot 3\).

For numbers: a whole number \(a>0\) is a factor of a number \(b\), if \(\frac{b}{a}\) is also a whole number.

For example, the factors of \(24\) are \(1,2,3,6,8,12,24\).

\(24 = 2\cdot 12 = 3\cdot 8 = 4\cdot 6 = 2\cdot 2\cdot 6 = 3\cdot2\cdot 4 = 2\cdot 2\cdot 2\cdot 3 = 2^3\cdot 3\).

Factors of \(29\) are \(1\) and \(29\).

Factors of a whole number \(a\) always include \(1\) and \(a\).

A positive whole number \(p\neq1\) is called a prime number if it does not have factors other than \(1\) and \(p\).

First prime numbers: \(2,3,5,7,11,13,17,19,23,29,\dots\)

Example 3.5 Factorise \(27+81\).

Solution. \(27+81 = 27(1+3)=27\cdot 4 = 3\cdot3\cdot3\cdot2\cdot2\).

Example 3.6 Factorise \(x^3-3x^2y\).

Solution. \(x^3-3x^2y = x^2(x-3y)\).

Example 3.7 Factorise \(6ax+3ay+2bx+by\).

Solution. \((6ax+3ay)+(2bx+by) = 3a(2x+y)+b(2x+y)\) \(=(3a+b)(2x+y)\).

Example 3.8 Factorise \(3x^2-5xy-2y^2\).

Solution. Note: \(-5=1\cdot 1-2\cdot 3\). So \(3x^2-5xy-2y^2\) \(=3x^2- 6xy + xy-2y^2\) \(=3x(x-2y)+y(x-2y)\) \(=(3x+y)(x-2y)\).

Example 3.9 Factorise \(25x^2y^2-30xy+9\).

Solution. \(25x^2y^2-30xy+9\) \(=25x^2y^2 -15xy -15xy +9\) \(=5xy(5xy-3)-3(5xy-3)\) \(=(5xy-3)(5xy-3)\).

Example 3.10 Factorise \(10a^2+ab-21b^2\).

Solution. \(10a^2+ab-21b^2\) \(=10a^2 +15ab - 14ab -21b^2\) \(=(2a+3b)(5a-7b)\).

Notes:

• It is not always possible to sensibly factorise a given expression.
• Positive whole numbers can be always factorised (uniquely in the appropriate sense) as a product of prime numbers.
• Useful formula: \(a^2-b^2=(a-b)(a+b)\).