smc-4696-10-Expand

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) ---

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\(\text{See worked solution}\)

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

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

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

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

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)

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\(x^2+xy\)

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

Algebraic Techniques, SM-Bank 136 MC

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

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

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\(B\)

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

\(\Rightarrow B\)

Algebraic Techniques, SM-Bank 135 MC

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

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

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\(D\)

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

\(\Rightarrow D\)

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)

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\(40x+xy\)

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

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)

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\(x^2+xy\)

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

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)

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\(12+4b\)

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

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)

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\(2y+12\)

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

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)

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\(10x+5\)

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

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)

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\(6b-10\)

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

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)

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\(2x+8\)

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

Algebraic Techniques, SM-Bank 127

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

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\(7+10x\)

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

Algebraic Techniques, SM-Bank 126

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

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\(11x+9\)

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

Algebraic Techniques, SM-Bank 125

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

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\(xy+10x\)

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

Algebraic Techniques, SM-Bank 124

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

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\(9m+4\)

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

Algebraic Techniques, SM-Bank 123

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

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\(9x^2-18xy\)

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

Algebraic Techniques, SM-Bank 122

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

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\(3b-b^2\)

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

Algebraic Techniques, SM-Bank 121

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

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\(x^2+2x\)

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

Algebraic Techniques, SM-Bank 120

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

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\(-a+8\)

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

Algebraic Techniques, SM-Bank 119

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

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\(-10y-40\)

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

Algebraic Techniques, SM-Bank 118

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

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\(4m-12\)

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

Algebraic Techniques, SM-Bank 117

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

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\(3x+3\)

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