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Övningar 2.1

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Versionen från 16 juli 2007 kl. 08.00 (redigera)
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(Ny sida: __NOTOC__ ==Övning 2.1:1== <div class="ovning"> Utveckla <table width="100%" cellspacing="10px"> <tr align="left"> <td class="ntext">a)</td> <td class="ntext" width="33%">3x(x-1)</td> <t...)
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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==Övning 2.1:1==+'''Övning 2.1:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Utveckla Utveckla
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">3x(x-1)</td> <td class="ntext" width="33%">3x(x-1)</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">(1+x-x^2)xy</td> <td class="ntext" width="33%">(1+x-x^2)xy</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">-x^2(4-y^2)</td> <td class="ntext" width="33%">-x^2(4-y^2)</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right)</td> <td class="ntext" width="33%">x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right)</td>
-<td class="ntext">e) </td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp; </td>
<td class="ntext" width="33%">(x-7)^2</td> <td class="ntext" width="33%">(x-7)^2</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">(5+4y)^2</td> <td class="ntext" width="33%">(5+4y)^2</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">g)</td>+<td class="ntext">g)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">(y^2-3x^3)^2</td> <td class="ntext" width="33%">(y^2-3x^3)^2</td>
-<td class="ntext">h)</td>+<td class="ntext">h)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">(5x^3+3x^5)^2</td> <td class="ntext" width="33%">(5x^3+3x^5)^2</td>
</tr> </tr>
Rad 30: Rad 30:
</div> </div>
-==Övning 2.1:2==+'''Övning 2.1:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Utveckla och förenkla så långt som möjligt Utveckla och förenkla så långt som möjligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">(x-4)(x-5)-3x(2x-3)</td> <td class="ntext" width="50%">(x-4)(x-5)-3x(2x-3)</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">(1-5x)(1+15x)-3(2-5x)(2+5x)</td> <td class="ntext" width="50%">(1-5x)(1+15x)-3(2-5x)(2+5x)</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">(3x+4)^2-(3x-2)(3x-8)</td> <td class="ntext" width="50%">(3x+4)^2-(3x-2)(3x-8)</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">(3x^2+2)(3x^2-2)(9x^4+4)</td> <td class="ntext" width="50%">(3x^2+2)(3x^2-2)(9x^4+4)</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
-<td class="ntext" width="50%">(a+b)^2+(a-b)^2</td>+<td class="ntext" width="50%">$(a+b)&nbsp;&nbsp;&nbsp;^2+(a-b)&nbsp;&nbsp;&nbsp;^2$</td>
</tr> </tr>
<tr><td height="5px"/></tr> <tr><td height="5px"/></tr>
Rad 54: Rad 54:
</div> </div>
-==Övning 2.1:3==+'''Övning 2.1:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Faktorisera s&aring; l&aring;ngt som m&ouml;jligt Faktorisera s&aring; l&aring;ngt som m&ouml;jligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2-36</td> <td class="ntext" width="33%">x^2-36</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">5x^2-20</td> <td class="ntext" width="33%">5x^2-20</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2+6x+9</td> <td class="ntext" width="33%">x^2+6x+9</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2-10x+25</td> <td class="ntext" width="33%">x^2-10x+25</td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">18x-2x^3</td> <td class="ntext" width="33%">18x-2x^3</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">16x^2+8x+1</td> <td class="ntext" width="33%">16x^2+8x+1</td>
</tr> </tr>
Rad 79: Rad 79:
</div> </div>
-==Övning 2.1:4==+'''Övning 2.1:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Bestäm koefficienterna framför \,x\, och \,x^2\, när följande uttryck utvecklas Bestäm koefficienterna framför \,x\, och \,x^2\, när följande uttryck utvecklas
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">(x+2)(3x^2-x+5)</td> <td class="ntext" width="100%">(x+2)(3x^2-x+5)</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">(1+x+x^2+x^3)(2-x+x^2+x^4)</td> <td class="ntext" width="100%">(1+x+x^2+x^3)(2-x+x^2+x^4)</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">(x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)</td> <td class="ntext" width="100%">(x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)</td>
</tr> </tr>
Rad 99: Rad 99:
</div> </div>
-==Övning 2.1:5==+'''Övning 2.1:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Förenkla så långt som möjligt Förenkla så långt som möjligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x}</td> <td class="ntext" width="50%">\displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}</td> <td class="ntext" width="50%">\displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)}</td> <td class="ntext" width="50%">\displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}</td> <td class="ntext" width="50%">\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}</td>
</tr> </tr>
Rad 118: Rad 118:
</div> </div>
-==Övning 2.1:6==+'''Övning 2.1:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Förenkla så långt som möjligt Förenkla så långt som möjligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\left(x-y+\displaystyle\frac{x^2}{y-x}\right) \left(\displaystyle\frac{y}{2x-y}-1\right)</td> <td class="ntext" width="50%">\left(x-y+\displaystyle\frac{x^2}{y-x}\right) \left(\displaystyle\frac{y}{2x-y}-1\right)</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2</td> <td class="ntext" width="50%">\displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b}</td> <td class="ntext" width="50%">\displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}</td> <td class="ntext" width="50%">\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}</td>
</tr> </tr>
Rad 138: Rad 138:
</div> </div>
-==Övning 2.1:7==+'''Övning 2.1:7'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{2}{x+3}-\frac{2}{x+5}</td> <td class="ntext" width="33%">\displaystyle \frac{2}{x+3}-\frac{2}{x+5}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}</td> <td class="ntext" width="33%">x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}</td> <td class="ntext" width="33%">\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}</td>
</tr> </tr>
Rad 153: Rad 153:
</div> </div>
-==Övning 2.1:8==+'''Övning 2.1:8'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr> <tr>
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }</td> <td class="ntext" width="33%">\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}</td> <td class="ntext" width="33%">\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}</td> <td class="ntext" width="33%">\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}</td>
</tr> </tr>
Rad 169: Rad 169:
</table> </table>
</div> </div>
 +<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>

Nuvarande version

Övning 2.1:1

Utveckla

a)    3x(x-1) b)    (1+x-x^2)xy c)    -x^2(4-y^2)
d)    x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right) e)    (x-7)^2 f)    (5+4y)^2
g)    (y^2-3x^3)^2 h)    (5x^3+3x^5)^2

Övning 2.1:2

Utveckla och förenkla så långt som möjligt

a)    (x-4)(x-5)-3x(2x-3) b)    (1-5x)(1+15x)-3(2-5x)(2+5x)
c)    (3x+4)^2-(3x-2)(3x-8) d)    (3x^2+2)(3x^2-2)(9x^4+4)
e)    (a+b)   ^2+(a-b)   ^2

Övning 2.1:3

Faktorisera så långt som möjligt

a)    x^2-36 b)    5x^2-20 c)    x^2+6x+9
d)    x^2-10x+25 e)    18x-2x^3 f)    16x^2+8x+1

Övning 2.1:4

Bestäm koefficienterna framför \,x\, och \,x^2\, när följande uttryck utvecklas

a)    (x+2)(3x^2-x+5)
b)    (1+x+x^2+x^3)(2-x+x^2+x^4)
c)    (x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)

Övning 2.1:5

Förenkla så långt som möjligt

a)    \displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x} b)    \displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}
c)    \displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)} d)    \displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}

Övning 2.1:6

Förenkla så långt som möjligt

a)    \left(x-y+\displaystyle\frac{x^2}{y-x}\right) \left(\displaystyle\frac{y}{2x-y}-1\right) b)    \displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2
c)    \displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b} d)    \displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}

Övning 2.1:7

Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck

a)    \displaystyle \frac{2}{x+3}-\frac{2}{x+5} b)    x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2} c)    \displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}

Övning 2.1:8

Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck

a)    \displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ } b)    \displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}} c)    \displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}


































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