2.1 Algebraic expressions - Revision history

Tek: MGN --> LCD

Tek — Tue, 02 Nov 2010 11:28:45 GMT

MGN --> LCD

←Older revision		Revision as of 11:28, 2 November 2010
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	<li><math>\frac{1}{x-2} - \frac{4}{x^2-4}		<li><math>\frac{1}{x-2} - \frac{4}{x^2-4}
	= \frac{1}{x-2} - \frac{4}{(x+2)(x-2)}		= \frac{1}{x-2} - \frac{4}{(x+2)(x-2)}
-	= \left\{\,\mbox{~~MGN~~}	+	= \left\{\,\mbox{LCD}
	= (x+2)(x-2)\,\right\}</math><br/><br/>		= (x+2)(x-2)\,\right\}</math><br/><br/>
	<math>\phantom{\frac{1}{x-2} - \frac{4}{x^2-4}}{}		<math>\phantom{\frac{1}{x-2} - \frac{4}{x^2-4}}{}

Eakc07: Minor grammatical changes

Eakc07 — Fri, 31 Oct 2008 16:44:07 GMT

Minor grammatical changes

←Older revision		Revision as of 16:44, 31 October 2008
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	'''Learning outcomes:'''		'''Learning outcomes:'''

-	After this section, you will have learned how to:	+	After this section you will have learned how to:

	*Simplify complex algebraic expressions.		*Simplify complex algebraic expressions.
-	*Factorise expressions ~~using~~ squaring rules and the difference of two squares rule.	+	*Factorise expressions with the use of squaring rules and the difference of two squares rule.
-	*Expand expressions ~~using~~ squaring rules and the difference of two squares rule.	+	*Expand expressions with the use of squaring rules and the difference of two squares rule.
	}}		}}

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	</div>		</div>

-	Using the distributive law, we can also see how to tackle	+	Using the distributive law we can also see how to tackle
	a minus sign in front of a bracketed expression.		a minus sign in front of a bracketed expression.
-	The rule says that ~~a bracket with~~ a minus sign ~~in front~~ can be	+	The rule says that a minus sign infront of a bracket can be
-	eliminated if all the terms inside the brackets, switch signs.	+	eliminated if all the terms inside the brackets switch signs.

	<div class="exempel">		<div class="exempel">
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	</div>		</div>

-	If the distributive law is applied in reverse we say we ~~“factor”~~ the expression.	+	If the distributive law is applied in reverse we say we “factorise” the expression.
-	One ~~often~~ would like to to factorise out ~~as large a~~ numerical factor ~~as possible~~.	+	One would often like to factorise out the largest possible numerical factor.

	<div class="exempel">		<div class="exempel">
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	== Squaring rules ==		== Squaring rules ==

-	~~The~~ distributive law ~~occasionally~~ has to be used repeatedly to deal with larger expressions.	+	On occasions the distributive law has to be used repeatedly to deal with larger expressions.
	If we consider		If we consider

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	</div>		</div>

-	These formulas are called the first and second squaring rules	+	These formulas are called the first and second squaring rules.

	<div class="exempel">		<div class="exempel">
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	= \dots</math>}}		= \dots</math>}}

-	The opposite of this, is cancellation~~, where~~ we delete factors that the numerator and denominator have in common	+	The opposite of this is cancellation. Here we delete factors that the numerator and denominator have in common
	{{Displayed math\|\|<math>\frac{(x+2)(x+3)(x+4)}{(x+1)(x+3)(x+4) }		{{Displayed math\|\|<math>\frac{(x+2)(x+3)(x+4)}{(x+1)(x+3)(x+4) }
	= \frac{(x+2)(x+4)}{(x+1)(x+4)}		= \frac{(x+2)(x+4)}{(x+1)(x+4)}
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	</div>		</div>

-	When fractional expressions are added or subtracted, they may need to be converted so that they have the same denominator ~~before~~ the numerators ~~can~~ be combined together,	+	When fractional expressions are added or subtracted they may need to be converted so that they have the same denominator. Only then can the numerators be combined together.


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	= \frac{-1}{x(x-1)} \; \mbox{.}</math>}}		= \frac{-1}{x(x-1)} \; \mbox{.}</math>}}

-	One normally tries to convert the fractions by multiplying the numerators and denominators by minimal factors to facilitate the calculations. The lowest common denominator (LCD) is the common denominator which contains the least number of factors.	+	One normally tries to convert the fractions by multiplying the numerators and denominators by minimal factors to facilitate the calculations. The lowest common denominator (LCD) is the common denominator which contains the least number of factors.

	<div class="exempel">		<div class="exempel">
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	</div>		</div>

-	To simplify large expressions, it is often necessary to both cancel factors and multiply numerators and denominators by factors. As cancellation implies that we have performed factorisations, it is obvious we should try to keep expressions (such as the denominator) factorised and not expand something that we will later need to factorise.	+	To simplify large expressions it is often necessary to both cancel factors and multiply numerators and denominators by factors. As cancellation implies that we have performed factorisations it is obvious we should try to keep expressions (such as the denominator) factorised and not expand something that we will later need to factorise.

Jdinglis at 13:58, 26 October 2008

Jdinglis — Sun, 26 Oct 2008 13:58:36 GMT

←Older revision		Revision as of 13:58, 26 October 2008
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-	'''Keep in mind that: '''	+	'''Keep in mind that...'''

-	~~Be careful.~~ If you make a mistake somewhere the rest of the calculation will be wrong.	+	If you make a mistake somewhere the rest of the calculation will be wrong, so be careful!

	Use many intermediate steps. If you are unsure of a calculation do it in many small steps rather than one big step.		Use many intermediate steps. If you are unsure of a calculation do it in many small steps rather than one big step.

Tekbot: Robot: Automated text replacement (-{{:2.1 - Figur - Distributiva lagen två gånger}} +{{:2.1 - Figure - The distributive law twice}})

Tekbot — Fri, 12 Sep 2008 12:19:33 GMT

Robot: Automated text replacement (-{{:2.1 - Figur - Distributiva lagen två gånger}} +{{:2.1 - Figure - The distributive law twice}})

←Older revision		Revision as of 12:19, 12 September 2008
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	A mnemonic for this formula is:		A mnemonic for this formula is:

-	<center>{{:2.1 - ~~Figur~~ - ~~Distributiva lagen två gånger~~}}</center>	+	<center>{{:2.1 - Figure - The distributive law twice}}</center>

	<div class="exempel">		<div class="exempel">

Tekbot: Robot: Automated text replacement (-{{:2.1 - Figur - Distributiva lagen}} +{{:2.1 - Figure - The distributive law}})

Tekbot — Fri, 12 Sep 2008 12:18:27 GMT

Robot: Automated text replacement (-{{:2.1 - Figur - Distributiva lagen}} +{{:2.1 - Figure - The distributive law}})

←Older revision		Revision as of 12:18, 12 September 2008
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	The distributive law specifies how to multiply a bracketed expression by a factor.		The distributive law specifies how to multiply a bracketed expression by a factor.

-	<center>{{:2.1 - ~~Figur~~ - ~~Distributiva lagen~~}}</center>	+	<center>{{:2.1 - Figure - The distributive law}}</center>

	<div class="exempel">		<div class="exempel">

Tekbot: Robot: Automated text replacement (-{{Fristående formel +{{Displayed math)

Tekbot — Wed, 10 Sep 2008 13:54:19 GMT

Robot: Automated text replacement (-{{Fristående formel +{{Displayed math)

←Older revision		Revision as of 13:54, 10 September 2008
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	If we consider		If we consider

-	{{~~Fristående formel~~\|\|<math>(a+b)(c+d)</math>}}	+	{{Displayed math\|\|<math>(a+b)(c+d)</math>}}

	and regard <math>a+b</math> as a factor that multiplies the bracketed expression <math>(c+d)</math> we get		and regard <math>a+b</math> as a factor that multiplies the bracketed expression <math>(c+d)</math> we get

-	{{~~Fristående formel~~\|\|<math>\eqalign{	+	{{Displayed math\|\|<math>\eqalign{
	\bbox[#AAEEFF,0pt]{\phantom{(a+b)}}\,(c+d)		\bbox[#AAEEFF,0pt]{\phantom{(a+b)}}\,(c+d)
	&= \bbox[#AAEEFF,0pt]{\phantom{(a+b)}}\,c		&= \bbox[#AAEEFF,0pt]{\phantom{(a+b)}}\,c
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	Then the <math>c</math> and the <math>d</math> are multiplied into their respective brackets,		Then the <math>c</math> and the <math>d</math> are multiplied into their respective brackets,

-	{{~~Fristående formel~~\|\|<math>(a+b)c + (a+b)d = ac + bc + ad + bd \, \mbox{.}</math>}}	+	{{Displayed math\|\|<math>(a+b)c + (a+b)d = ac + bc + ad + bd \, \mbox{.}</math>}}

	A mnemonic for this formula is:		A mnemonic for this formula is:
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	<div class="regel">		<div class="regel">
	'''Squaring rules '''		'''Squaring rules '''
-	{{~~Fristående formel~~\|\|<math>(a+b)^2 = a^2 +2ab + b^2</math>}}	+	{{Displayed math\|\|<math>(a+b)^2 = a^2 +2ab + b^2</math>}}
-	{{~~Fristående formel~~\|\|<math>(a-b)^2 = a^2 -2ab + b^2</math>}}	+	{{Displayed math\|\|<math>(a-b)^2 = a^2 -2ab + b^2</math>}}
	</div>		</div>

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	<div class="regel">		<div class="regel">
	'''Difference of two squares:'''		'''Difference of two squares:'''
-	{{~~Fristående formel~~\|\|<math>(a+b)(a-b) = a^2 -b^2</math>}}	+	{{Displayed math\|\|<math>(a+b)(a-b) = a^2 -b^2</math>}}
	</div>		</div>

	This formula can be obtained directly by expanding the left hand side		This formula can be obtained directly by expanding the left hand side

-	{{~~Fristående formel~~\|\|<math>(a+b)(a-b)	+	{{Displayed math\|\|<math>(a+b)(a-b)
	= a \cdot a + a\cdot (-b) + b\cdot a + b \cdot (-b)		= a \cdot a + a\cdot (-b) + b\cdot a + b \cdot (-b)
	= a^2 -ab+ab-b^2		= a^2 -ab+ab-b^2
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	<div class="regel">		<div class="regel">
-	{{~~Fristående formel~~\|\|<math> \frac{a}{b} \cdot \frac{c}{d}	+	{{Displayed math\|\|<math> \frac{a}{b} \cdot \frac{c}{d}
	= \frac{a\cdot c}{b\cdot d}		= \frac{a\cdot c}{b\cdot d}
	\quad \mbox{and} \quad		\quad \mbox{and} \quad
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	A fractional expression can have its numerator and denominator multiplied by the same factor		A fractional expression can have its numerator and denominator multiplied by the same factor

-	{{~~Fristående formel~~\|\|<math>\frac{x+2}{x+1}	+	{{Displayed math\|\|<math>\frac{x+2}{x+1}
	= \frac{(x+2)(x+3)}{(x+1)(x+3)}		= \frac{(x+2)(x+3)}{(x+1)(x+3)}
	= \frac{(x+2)(x+3)(x+4)}{(x+1)(x+3)(x+4)}		= \frac{(x+2)(x+3)(x+4)}{(x+1)(x+3)(x+4)}
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	The opposite of this, is cancellation, where we delete factors that the numerator and denominator have in common		The opposite of this, is cancellation, where we delete factors that the numerator and denominator have in common
-	{{~~Fristående formel~~\|\|<math>\frac{(x+2)(x+3)(x+4)}{(x+1)(x+3)(x+4) }	+	{{Displayed math\|\|<math>\frac{(x+2)(x+3)(x+4)}{(x+1)(x+3)(x+4) }
	= \frac{(x+2)(x+4)}{(x+1)(x+4)}		= \frac{(x+2)(x+4)}{(x+1)(x+4)}
	= \frac{x+2}{x+1} \mbox{.}</math>}}		= \frac{x+2}{x+1} \mbox{.}</math>}}
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-	{{~~Fristående formel~~\|\|<math>\frac{1}{x} - \frac{1}{x-1}	+	{{Displayed math\|\|<math>\frac{1}{x} - \frac{1}{x-1}
	= \frac{1}{x} \cdot \frac{x-1}{x-1} - \frac{1}{x-1} \cdot \frac{x}{x}		= \frac{1}{x} \cdot \frac{x-1}{x-1} - \frac{1}{x-1} \cdot \frac{x}{x}
	= \frac{x-1}{x(x-1)} - \frac{x}{x(x-1)}		= \frac{x-1}{x(x-1)} - \frac{x}{x(x-1)}
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	= (x+1)(x+2)</math> <br><br>		= (x+1)(x+2)</math> <br><br>
	Convert the first term using <math>(x+2)</math> and the second term using <math>(x+1)</math>		Convert the first term using <math>(x+2)</math> and the second term using <math>(x+1)</math>
-	{{~~Fristående formel~~\|\|<math>\begin{align*}	+	{{Displayed math\|\|<math>\begin{align*}
	\frac{1}{x+1} + \frac{1}{x+2}		\frac{1}{x+1} + \frac{1}{x+2}
	&= \frac{x+2}{(x+1)(x+2)} + \frac{x+1}{(x+2)(x+1)}\\[4pt]		&= \frac{x+2}{(x+1)(x+2)} + \frac{x+1}{(x+2)(x+1)}\\[4pt]
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	= x^2</math><br><br>		= x^2</math><br><br>
	We only need to convert the first term to get a common denominator		We only need to convert the first term to get a common denominator
-	{{~~Fristående formel~~\|\|<math>\frac{1}{x} + \frac{1}{x^2}	+	{{Displayed math\|\|<math>\frac{1}{x} + \frac{1}{x^2}
	= \frac{x}{x^2} + \frac{1}{x^2}		= \frac{x}{x^2} + \frac{1}{x^2}
	= \frac{x+1}{x^2}\,\mbox{.}</math>}}</li>		= \frac{x+1}{x^2}\,\mbox{.}</math>}}</li>
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	\text{LCD}= x^2(x+1)^2(x+2)</math><br><br>		\text{LCD}= x^2(x+1)^2(x+2)</math><br><br>
	The first term is converted using <math>x(x+2)</math> while the other term is converted using <math>(x+1)^2</math>		The first term is converted using <math>x(x+2)</math> while the other term is converted using <math>(x+1)^2</math>
-	{{~~Fristående formel~~\|\|<math>\begin{align*}	+	{{Displayed math\|\|<math>\begin{align*}
	\frac{1}{x(x+1)^2} - \frac{1}{x^2(x+2)}		\frac{1}{x(x+1)^2} - \frac{1}{x^2(x+2)}
	&= \frac{x(x+2)}{x^2(x+1)^2(x+2)}		&= \frac{x(x+2)}{x^2(x+1)^2(x+2)}
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	\text{LCD}=x(x-1)(x+1)</math><br><br>		\text{LCD}=x(x-1)(x+1)</math><br><br>
	We must convert all the terms so that they have the common denominator <math>x(x-1)(x+1)</math>		We must convert all the terms so that they have the common denominator <math>x(x-1)(x+1)</math>
-	{{~~Fristående formel~~\|\|<math>\begin{align*}	+	{{Displayed math\|\|<math>\begin{align*}
	\frac{x}{x+1} - \frac{1}{x(x-1)} -1		\frac{x}{x+1} - \frac{1}{x(x-1)} -1
	&= \frac{x^2(x-1)}{x(x-1)(x+1)} - \frac{x+1}{x(x-1)(x+1)}		&= \frac{x^2(x-1)}{x(x-1)(x+1)} - \frac{x+1}{x(x-1)(x+1)}

Tekbot: Robot: Automated text replacement (-{{Vald flik +{{Selected tab)

Tekbot — Wed, 10 Sep 2008 13:49:25 GMT

Robot: Automated text replacement (-{{Vald flik +{{Selected tab)

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-	{{~~Vald flik~~\|[[2.1 Algebraic expressions\|Theory]]}}	+	{{Selected tab\|[[2.1 Algebraic expressions\|Theory]]}}
	{{Not selected tab\|[[2.1 Exercises\|Exercises]]}}		{{Not selected tab\|[[2.1 Exercises\|Exercises]]}}
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Tekbot: Robot: Automated text replacement (-{{Ej vald flik +{{Not selected tab)

Tekbot — Wed, 10 Sep 2008 13:44:02 GMT

Robot: Automated text replacement (-{{Ej vald flik +{{Not selected tab)

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	{{Vald flik\|[[2.1 Algebraic expressions\|Theory]]}}		{{Vald flik\|[[2.1 Algebraic expressions\|Theory]]}}
-	{{~~Ej vald flik~~\|[[2.1 Exercises\|Exercises]]}}	+	{{Not selected tab\|[[2.1 Exercises\|Exercises]]}}
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Tekbot: Robot: Automated text replacement (-2.1 Algebraiska uttryck +2.1 Algebraic expressions)

Tekbot — Wed, 10 Sep 2008 13:13:51 GMT

Robot: Automated text replacement (-2.1 Algebraiska uttryck +2.1 Algebraic expressions)

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-	{{Vald flik\|[[2.1 ~~Algebraiska uttryck~~\|Theory]]}}	+	{{Vald flik\|[[2.1 Algebraic expressions\|Theory]]}}
	{{Ej vald flik\|[[2.1 Exercises\|Exercises]]}}		{{Ej vald flik\|[[2.1 Exercises\|Exercises]]}}
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Tekbot: Robot: Automated text replacement (-Övningar +Exercises)

Tekbot — Wed, 10 Sep 2008 10:38:01 GMT

Robot: Automated text replacement (-Övningar +Exercises)

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-	{{Ej vald flik\|[[2.1 ~~Övningar~~\|Exercises]]}}	+	{{Ej vald flik\|[[2.1 Exercises\|Exercises]]}}
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-	{{~~Vald flik~~\|[[2.1 Algebraic expressions\|Theory]]}}	+	{{Selected tab\|[[2.1 Algebraic expressions\|Theory]]}}
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	{{Vald flik\|[[2.1 Algebraic expressions\|Theory]]}}		{{Vald flik\|[[2.1 Algebraic expressions\|Theory]]}}
-	{{~~Ej vald flik~~\|[[2.1 Exercises\|Exercises]]}}	+	{{Not selected tab\|[[2.1 Exercises\|Exercises]]}}
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