smc-4696-20-Factorise

Algebraic Techniques, SM-Bank 156

Fully factorise \(24p^2qr^3+18p^3q^2+6p^2qr^2\). (2 marks)

Show Answers Only

\(6p^2q(4r^3+3pq+r^2)\)

Show Worked Solution

\(24p^2qr^3+18p^3q^2+6p^2qr^2\)	\(=6p^2q\times 4r^3+6p^2q\times 3pq+6p^2q\times r^2\ \ \ \ (\text{HCF}=6p^2q)\)
	\(=6p^2q(4r^3+3pq+r^2)\)

Algebraic Techniques, SM-Bank 155

Fully factorise \(2mn^2+6m^2n-8mn\). (2 marks)

Show Answers Only

\(2mn(n+3m-4)\)

Show Worked Solution

\(2mn^2+6m^2n-8mn\)	\(=2mn\times n+2mn\times 3m+2mn\times (-4)\ \ \ \ (\text{HCF}=2mn)\)
	\(=2mn(n+3m-4)\)

Algebraic Techniques, SM-Bank 154

A rectangle has an area of \(2pq^2+4p^2q\). By factorising the expression, find the length of the rectangle if the width is \(2pq\). (2 marks)

Show Answers Only

\(q+2p\)

Show Worked Solution

\(2pq^2+4pq^2\)	\(=2pq\times q+2pq\times 2p\ \ \ \ (\text{HCF}=2pq)\)
	\(=2pq(q+2p)\)

\(\therefore\ \text{Length of rectangle}=q+2p\)

Algebraic Techniques, SM-Bank 153

A rectangle has an area of \(9a^2-6a\). By factorising the expression, find the width of the rectangle if the length is \(3a\). (2 marks)

Show Answers Only

\(3a-2\)

Show Worked Solution

\(9a^2-6a\)	\(=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)\)
	\(=3a(3a-2)\)

\(\therefore\ \text{Width of rectangle}=3a-2\)

Algebraic Techniques, SM-Bank 152

A rectangle has an area of \(9a^2-6a\). By factorising the expression, find the width of the rectangle if the length is \(3a\). (2 marks)

Show Answers Only

\(3a-2\)

Show Worked Solution

\(9a^2-6a\)	\(=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)\)
	\(=3a(3a-2)\)

\(\therefore\ \text{Width of rectangle}=3a-2\)

Algebraic Techniques, SM-Bank 151

Fully factorise the following:

\(4x^2-2x\) (2 marks)

--- 2 WORK AREA LINES (style=lined) ---
\(-6ab+3a\) (2 marks)

--- 2 WORK AREA LINES (style=lined) ---
\(-5q-10q^2\) (2 marks)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. \(2x(2x-1)\)

b. \(-3a(2b-1)\)

c. \(-5q(1+2q)\)

Show Worked Solution

a.	\(4x^2-2x\)	\(=2x\times 2x+2x\times (-1)\ \ \ \ (\text{HCF}=2x)\)
		\(=2x(2x-1)\)

b.	\(-6ab+3a\)	\(=-3a\times 2b-3a\times(-1)\ \ \ \ (\text{HCF}=-3a)\)
		\(=-3a(2b-1)\)

c.	\(-5q-10q^2\)	\(=-5q\times 1-5q\times 2q\ \ \ \ (\text{HCF}=-5q)\)
		\(=-5q(1+2q)\)

Algebraic Techniques, SM-Bank 150 MC

Which of the following is the correct factorisation of \(7y+14x\).

\(7(y+2x)\)
\(y(7+14x)\)
\(7(y+14x)\)
\(7(y+7x)\)

Show Answers Only

\(A\)

Show Worked Solution

\(7y+14x\)	\(=7\times y+7\times 2x\ \ \ \ (\text{HCF}=7)\)
	\(=7(y+2x)\)

\(\Rightarrow A\)

Algebraic Techniques, SM-Bank 149 MC

Which of the following is the correct factorisation of \(15m-20\).

\(15(m-20)\)
\(5(m-4)\)
\(5(3m-4)\)
\(5(3m-20)\)

Show Answers Only

\(C\)

Show Worked Solution

\(15m-20\)	\(=5\times 3m-5\times 4\ \ \ \ (\text{HCF}=5)\)
	\(=5(3m-4)\)

\(\Rightarrow C\)

Algebraic Techniques, SM-Bank 148 MC

Which of the following is the correct factorisation of \(3x-12\).

\(3(x-12)\)
\(x(3-12x)\)
\(3(x-9)\)
\(3(x-4)\)

Show Answers Only

\(D\)

Show Worked Solution

\(3x-12\)	\(=3\times x-3\times 4\ \ \ \ (\text{HCF}=3)\)
	\(=3(x-4)\)

\(\Rightarrow D\)

Algebraic Techniques, SM-Bank 147

State the highest common factor of \(36mn^2\ \text{and}\ 24m^2n\). (1 mark)

Show Answers Only

\(12mn\)

Show Worked Solution

\(\text{HCF of}\ 36\ \text{and}\ 24=12\)

\(\text{HCF of}\ mn^2\ \text{and}\ m^2n=mn\)

\(\therefore\ \text{HCF of}\ 36mn^2\ \text{and}\ 24m^2n=12mn\)

Algebraic Techniques, SM-Bank 146

State the highest common factor of \(27ab^2c\ \text{and}\ 3a^2b^2c\). (1 mark)

Show Answers Only

\(3ab^2c\)

Show Worked Solution

\(\text{HCF of}\ 27\ \text{and}\ 3=3\)

\(\text{HCF of}\ ab^2c\ \text{and}\ a^2b^2c=ab^2c\)

\(\therefore\ \text{HCF of}\ 27ab^2c\ \text{and}\ 3a^2b^2c=3ab^2c\)

Algebraic Techniques, SM-Bank 145

State the highest common factor of \(15xy\ \text{and}\ 20y\). (1 mark)

Show Answers Only

\(5y\)

Show Worked Solution

\(\text{HCF of}\ 15\ \text{and}\ 20=5\)

\(\text{HCF of}\ xy\ \text{and}\ y=y\)

\(\therefore\ \text{HCF of}\ 15xy\ \text{and}\ 20y=5y\)

Algebraic Techniques, SM-Bank 144

List all the factors of \(-6z\). (2 marks)

Show Answers Only

\( 1\ ,\ 2\ ,\ 3\ ,\ 6\ ,-z\ ,-2z \ ,-3z\ ,-6z\)

\( -1\ , -2\ , -3\ , -6\ ,\ z\ ,\ 2z \ ,\ 3z\ ,\ 6z\)

Show Worked Solution

\(\text{Listing factors in pairs}\)

\((1\ ,-6z) (2\ ,-3z) (3\ ,-2z) (6\ ,-z)\)

\((-1 ,\ 6z) (-2 , 3z) (-3 ,\ 2z) (-6 , z)\)

\(\therefore\ \text{Factors are:}\)

\( 1\ ,\ 2\ ,\ 3\ ,\ 6\ ,-z\ ,-2z \ ,-3z\ ,-6z\)

\( -1\ , -2\ , -3\ , -6\ ,\ z\ ,\ 2z \ ,\ 3z\ ,\ 6z\)

Algebraic Techniques, SM-Bank 143

List all the factors of \(-7q\). (2 marks)

Show Answers Only

\( 1\ ,\ 7\ ,-q\ ,-7q \ , -1\ , -7\ ,\ q\ ,\ 7q\)

Show Worked Solution

\(\text{Listing factors in pairs}\)

\(\longrightarrow (1\ ,-7q) (7\ ,-q) (-1 ,\ 7q) (-7 , q) \)

\(\therefore\ \text{Factors are:}\)

\( 1\ ,\ 7\ ,-q\ ,-7q \ , -1\ , -7\ ,\ q\ ,\ 7q\)

Algebraic Techniques, SM-Bank 142

List all the factors of \(-12y\). (2 marks)

Show Answers Only

\( 1\ ,\ 2\ ,\ 3\ ,\ 4\ ,\ 6\ ,\ 12\ ,-y\ ,-2y \ ,-3y\ ,-4y\ ,-6y\ ,-12y\)

\( -1\ , -2\ , -3\ , -4\ , -6\ , -12\ ,\ y\ ,\ 2y \ ,\ 3y\ ,\ 4y\ ,\ 6y\ ,\ 12y\)

Show Worked Solution

\(\text{Listing factors in pairs}\)

\((1\ ,-12y) (2\ ,-6y) (3\ ,-4y) (4\ ,-3y) (6\ ,-2y) (12\ ,-y)\)

\((-1 ,\ 12y) (-2 , 6y) (-3 , 4y) (-4 , 3y)(-6 , 2y) (-12 , y)\)

\(\therefore\ \text{Factors are:}\)

\( 1\ ,\ 2\ ,\ 3\ ,\ 4\ ,\ 6\ ,\ 12\ ,-y\ ,-2y \ ,-3y\ ,-4y\ ,-6y\ ,-12y\)

\( -1\ , -2\ , -3\ , -4\ , -6\ , -12\ ,\ y\ ,\ 2y \ ,\ 3y\ ,\ 4y\ ,\ 6y\ ,\ 12y\)

Algebraic Techniques, SM-Bank 141

List all the factors of \(10m\). (2 marks)

Show Answers Only

\( 1\ ,\ 2\ ,\ 5\ ,\ 10\ ,\ m\ ,\ 2m \ ,\ 5m\ ,\ 10m\)

\(-1 , -2 , -5 , -10 , -m , -2m , -5m , -10m\)

Show Worked Solution

\(\text{Listing factors in pairs}\)

\((1\ ,\ 10m) (2\ ,\ 5m) (5\ ,\ 2m) (10\ ,\ m) (-1 ,-10m) (-2 , -5m) (-5 , -2m) (-10,-m)\)

\(\therefore\ \text{Factors are:}\)

\( 1\ ,\ 2\ ,\ 5\ ,\ 10\ ,\ m\ ,\ 2m \ ,\ 5m\ ,\ 10m\)

\(-1 , -2 , -5 , -10 , -m , -2m , -5m , -10m\)

Algebraic Techniques, SM-Bank 140

List all the factors of \(2x\). (2 marks)

Show Answers Only

\(\ 1\ ,\ 2\ ,\ x\ ,\ 2x \ , -1\ , -2 \ , -x\ , -2x\)

Show Worked Solution

\(\text{Listing factors in pairs}\)

\(\longrightarrow (1\ ,\ 2x)\ \ (2\ ,\ x)\ \ (-1\ ,-2x)\ \ (-2\ ,-x)\)

\(\therefore \text{Factors are:}\ 1\ ,\ 2\ ,\ x\ ,\ 2x \ , -1\ , -2 \ , -x\ , -2x\)

Algebraic Techniques, SM-Bank 139

List all the factors of \(4b\). (2 marks)

Show Answers Only

\(\ 1\ ,\ 2\ ,\ 4\ ,\ b\ ,\ 2b ,\ 4b\ , -1\ , -2\ , -4\ , -b\ , -2b\ , -4b\)

Show Worked Solution

\(\text{Listing factors in pairs}\)

\((1\ ,\ 4b)\ \ (2\ ,\ 2b)\ \ (4\ ,\ b)\)

\((-1\ ,-4b)\ \ (-2\ ,-2b)\ \ (-4\ ,-b)\)

\(\therefore\ \text{Factors are:}\)

\(\ 1\ ,\ 2\ ,\ 4\ ,\ b\ ,\ 2b ,\ 4b ,\ -1\ , -2\ , -4\ , -b\ , -2b\ , -4b\)