Expand and 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

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

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

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

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

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

Algebraic Techniques, SM-Bank 138

Use the rectangle below to prove that \((a+b)^2=a^2+2ab+b^2\). (3 marks)

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

Show Answers Only

\(\text{See worked solution}\)

Show Worked Solution

\(\text{Length of large rectangle}\)	\(=a+b\)
\(\text{Width of large rectangle}\)	\(=a+b\)

\(\text{Area of large rectangle}\)	\(=(a+b)\times(a+b)\)
	\(=(a+b)^2\)

\(\text{Adding 4 areas inside large rectangle}\)	\(=a^2+a\times b+a\times b+b^2\)
	\(=a^2+2ab+b^2\)

\(\therefore (a+b)^2=a^2+2ab+b^2\)

Algebraic Techniques, SM-Bank 137

The triangle below has a base of \((2+6\large m)\) and a perpendicular height of \(\large m\).

Write an algebraic expression and expand it to represent the area of the triangle. (2 marks)

Show Answers Only

\(x^2+xy\)

Show Worked Solution

\(\text{Area of Triangle}\)	\(=\dfrac{1}{2}\times \text{base} \times \text{height}\)
	\(=\dfrac{1}{2}\times \Big(2+6m\Big)\times m\)
	\(=\dfrac{m}{2}\Big(2+6m\Big)\)
	\(=\dfrac{m}{2}\times 2 +\dfrac{m}{2}\times 6m\)
	\(=m+3m^2\)

Algebraic Techniques, SM-Bank 136 MC

The expansion of \(-2q(q-8)\) is:

\(-2q+16\)
\(-2q^2+16q\)
\(-2q^2-16q\)
\(2q^2+16q\)

Show Answers Only

\(B\)

Show Worked Solution

\(-2q(q-8)\)	\(=-2q\times q-2q\times (-8)\)
	\(=-2q^2+16q\)

\(\Rightarrow B\)

Algebraic Techniques, SM-Bank 135 MC

The expansion of \(4x(5-y)\) is:

\(9x-4xy\)
\(4x-5xy\)
\(20-4xy\)
\(20x-4xy\)

Show Answers Only

\(D\)

Show Worked Solution

\(4x(5-y)\)	\(=4x\times 5+4x\times (-y)\)
	\(=20x-4xy\)

\(\Rightarrow D\)

Algebraic Techniques, SM-Bank 134

Jimmy works \(\large x\) hours every week and his hourly rate of pay is \((40+\large y)\).

Write an algebraic expression and expand it to represent Jimmy's weekly wage. (2 marks)

Show Answers Only

\(40x+xy\)

Show Worked Solution

\(x(40+y)\)	\(=x\times 40+x\times y\)
	\(=40x+xy\)

Algebraic Techniques, SM-Bank 133

The rectangle below has a length of \((\large x+y)\) and a width of \(\large x\).

Write an algebraic expression and expand it to represent the area of the rectangle. (2 marks)

Show Answers Only

\(x^2+xy\)

Show Worked Solution

\(x(x+y)\)	\(=x\times x+x\times y\)
	\(=x^2+xy\)

Algebraic Techniques, SM-Bank 132

The rectangle below has a length of \((3+\large b)\) and a width of 4.

Write an algebraic expression and expand it to represent the area of the rectangle. (2 marks)

Show Answers Only

\(12+4b\)

Show Worked Solution

\(4(3+b)\)	\(=4\times 3+4\times b\)
	\(=12+4b\)

Algebraic Techniques, SM-Bank 131

A number \(\large y\) is halved and 3 is added. The result is then multiplied by 4.

Write an algebraic expression and expand it to represent this information. (2 marks)

Show Answers Only

\(2y+12\)

Show Worked Solution

\(4\bigg(\dfrac{y}{2}+3\bigg)\)	\(=4\times \dfrac{y}{2}+4\times 3\)
	\(=\dfrac{4y}{2}+12\)
	\(=2y+12\)

Algebraic Techniques, SM-Bank 130

A number \(\large x\) is doubled and 1 is added. The result is then multiplied by 5.

Write an algebraic expression and expand it to represent this information. (2 marks)

Show Answers Only

\(10x+5\)

Show Worked Solution

\(5(2x+1)\)	\(=5\times 2x+5\times 1\)
	\(=10x+5\)

Algebraic Techniques, SM-Bank 129

A number \(b\) is tripled and \(5\) is subtracted. The result is then doubled.

Write an algebraic expression and expand it to represent this information. (2 marks)

Show Answers Only

\(6b-10\)

Show Worked Solution

\(2(3b-5)\)	\(=2\times 3b+2\times (-5)\)
	\(=6b-10\)

Algebraic Techniques, SM-Bank 128

A number \(x\) is added to \(4\) and the result is doubled.

Write an algebraic expression and expand it to represent this information. (2 marks)

Show Answers Only

\(2x+8\)

Show Worked Solution

\(2(x+4)\)	\(=2\times x+2\times 4\)
	\(=2x+8\)

Algebraic Techniques, SM-Bank 127

Expand and simplify the expression \(2(5-4x)+3(6x-1)\). (2 marks)

Show Answers Only

\(7+10x\)

Show Worked Solution

\(2(5-4x)+3(6x-1)\)	\(=2\times 5+2\times (-4x)+3\times 6x+3\times (-1)\)
	\(=10-8x+18x-3\)
	\(=7+10x\)

Algebraic Techniques, SM-Bank 126

Expand and simplify the expression \(3(x-1)+4(2x+3)\). (2 marks)

Show Answers Only

\(11x+9\)

Show Worked Solution

\(3(x-1)+4(2x+3)\)	\(=3\times x-3\times 1+4\times 2x+4\times 3\)
	\(=3x-3+8x+12\)
	\(=11x+9\)

Algebraic Techniques, SM-Bank 125

Expand and simplify the expression \(3xy-2x(y-5)\). (2 marks)

Show Answers Only

\(xy+10x\)

Show Worked Solution

\(3xy-2x(y-5)\)	\(=3xy-2x\times y-2x\times (-5)\)
	\(=3xy-2xy+10x\)
	\(=xy+10x\)

Algebraic Techniques, SM-Bank 124

Expand and simplify the expression \(2(m+2)+7m\). (2 marks)

Show Answers Only

\(9m+4\)

Show Worked Solution

\(2(m+2)+7m\)	\(=2\times m+2\times 2+7m\)
	\(=2m+4+7m\)
	\(=9m+4\)

Algebraic Techniques, SM-Bank 123

Use the distributive law to expand the expression \(9x(x-2y)\). (2 marks)

Show Answers Only

\(9x^2-18xy\)

Show Worked Solution

\(9x(x-2y)\)	\(=9x\times x-9x\times 2y\)
	\(=9x^2-18xy\)

Algebraic Techniques, SM-Bank 122

Use the distributive law to expand the expression \(b(3-b)\). (2 marks)

Show Answers Only

\(3b-b^2\)

Show Worked Solution

\(b(3-b)\)	\(=b\times 3-b\times b\)
	\(=3b-b^2\)

Algebraic Techniques, SM-Bank 121

Use the distributive law to expand the expression \(x(x+2)\). (2 marks)

Show Answers Only

\(x^2+2x\)

Show Worked Solution

\(x(x+2)\)	\(=x\times x+x\times 2\)
	\(=x^2+2x\)

Algebraic Techniques, SM-Bank 120

Use the distributive law to expand the expression \(-(a-8)\). (2 marks)

Show Answers Only

\(-a+8\)

Show Worked Solution

\(-(a-8)\)	\(=-1\times a-(-1)\times 8\)
	\(=-a+8\)

Algebraic Techniques, SM-Bank 119

Use the distributive law to expand the expression \(-10(y+4)\). (2 marks)

Show Answers Only

\(-10y-40\)

Show Worked Solution

\(-10(y+4)\)	\(=-10\times y+(-10)\times 4\)
	\(=-10y-40\)

Algebraic Techniques, SM-Bank 118

Use the distributive law to expand the expression \(4(m-3)\). (2 marks)

Show Answers Only

\(4m-12\)

Show Worked Solution

\(4(m-3)\)	\(=4\times m-4\times 3\)
	\(=4m-12\)

Algebraic Techniques, SM-Bank 117

Use the distributive law to expand the expression \(3(x+1)\). (2 marks)

Show Answers Only

\(3x+3\)

Show Worked Solution

\(3(x+1)\)	\(=3\times x+3\times 1\)
	\(=3x+3\)