Övningar 2.1
Sommarmatte 1
(Skillnad mellan versioner)
| Versionen från 16 juli 2007 kl. 08.00 (redigera) KTH.SE:u1zpa8nw (Diskussion | bidrag) (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...) ← Gå till föregående ändring |
Versionen från 16 juli 2007 kl. 10.44 (redigera) (ogör) KTH.SE:u1zpa8nw (Diskussion | bidrag) Gå till nästa ändring → |
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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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </td> |
| - | <td class="ntext" width="50%">$(a+b)^2+(a-b)^2$</td> | + | <td class="ntext" width="50%">$(a+b) ^2+(a-b) ^2$</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 så långt som möjligt | Faktorisera 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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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) </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> | ||
Versionen från 16 juli 2007 kl. 10.44
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) | $\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}$ |
