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2.1 Exercises

From Förberedande kurs i matematik 1

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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
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{{Mall:Ej vald flik|[[1.2 Bråkräkning|Teori]]}}
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{{Not selected tab|[[2.1 Algebraic expressions|Theory]]}}
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{{Mall:Vald flik|[[1.2 Övningar|Övningar]]}}
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{{Selected tab|[[2.1 Exercises|Exercises]]}}
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===Övning 2.1:1===
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===Exercise 2.1:1===
<div class="ovning">
<div class="ovning">
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Utveckla
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Expand
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
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||<math> (5x^3+3x^5)^2</math>
||<math> (5x^3+3x^5)^2</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 2.1:1|Lösning a|Lösning 2.1:1a|Lösning b|Lösning 2.1:1b|Lösning c|Lösning 2.1:1c|Lösning d|Lösning 2.1:1d|Lösning e|Lösning 2.1:1e|Lösning f|Lösning 2.1:1f|Lösning g|Lösning 2.1:1g|Lösning h|Lösning 2.1:1h}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:1|Solution a|Solution 2.1:1a|Solution b|Solution 2.1:1b|Solution c|Solution 2.1:1c|Solution d|Solution 2.1:1d|Solution e|Solution 2.1:1e|Solution f|Solution 2.1:1f|Solution g|Solution 2.1:1g|Solution h|Solution 2.1:1h}}
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===Övning 2.1:2===
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===Exercise 2.1:2===
<div class="ovning">
<div class="ovning">
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Utveckla
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Expand
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
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||<math> (a+b)^2+(a-b)^2</math>
||<math> (a+b)^2+(a-b)^2</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.1:2|Lösning a|Lösning 2.1:2a|Lösning b|Lösning 2.1:2b|Lösning c|Lösning 2.1:2c|Lösning d|Lösning 2.1:2d|Lösning e|Lösning 2.1:2e}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:2|Solution a|Solution 2.1:2a|Solution b|Solution 2.1:2b|Solution c|Solution 2.1:2c|Solution d|Solution 2.1:2d|Solution e|Solution 2.1:2e}}
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===Övning 2.1:3===
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===Exercise 2.1:3===
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<div class="ovning">
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Faktorisera s&aring; l&aring;ngt som m&ouml;jligt
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Factorise and simplify as much as possible
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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||<math> 16x^2+8x+1</math>
||<math> 16x^2+8x+1</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 2.1:3|Lösning a|Lösning 2.1:3a|Lösning b|Lösning 2.1:3b|Lösning c|Lösning 2.1:3c|Lösning d|Lösning 2.1:3d|Lösning e|Lösning 2.1:3e|Lösning f|Lösning 2.1:3f}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:3|Solution a|Solution 2.1:3a|Solution b|Solution 2.1:3b|Solution c|Solution 2.1:3c|Solution d|Solution 2.1:3d|Solution e|Solution 2.1:3e|Solution f|Solution 2.1:3f}}
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===Övning 2.1:4===
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===Exercise 2.1:4===
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<div class="ovning">
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Bestäm koefficienterna framför <math>\,x\,</math> och <math>\,x^2\</math> när följande uttryck utvecklas
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Determine the coefficients in front of <math>\,x\,</math> and <math>\,x^2\</math> when the following expressions are expanded out.
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
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|a)
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</div>{{#NAVCONTENT:Svar|Svar 2.1:4|Lösning a|Lösning 2.1:4a|Lösning b|Lösning 2.1:4b|Lösning c|Lösning 2.1:4c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:4|Solution a|Solution 2.1:4a|Solution b|Solution 2.1:4b|Solution c|Solution 2.1:4c}}
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===Övning 2.1:5===
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===Exercise 2.1:5===
<div class="ovning">
<div class="ovning">
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Förenkla så långt som möjligt
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Simplify as much as possible
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|| <math>\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}</math>
|| <math>\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.1:5|Lösning a|Lösning 2.1:5a|Lösning b|Lösning 2.1:5b|Lösning c|Lösning 2.1:5c|Lösning d|Lösning 2.1:5d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:5|Solution a|Solution 2.1:5a|Solution b|Solution 2.1:5b|Solution c|Solution 2.1:5c|Solution d|Solution 2.1:5d}}
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===Övning 2.1:6===
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===Exercise 2.1:6===
<div class="ovning">
<div class="ovning">
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Förenkla så långt som möjligt
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Simplify as much as possible
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
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|a)
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|| <math>\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}</math>
|| <math>\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.1:6|Lösning a|Lösning 2.1:6a|Lösning b|Lösning 2.1:6b|Lösning c|Lösning 2.1:6c|Lösning d|Lösning 2.1:6d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:6|Solution a|Solution 2.1:6a|Solution b|Solution 2.1:6b|Solution c|Solution 2.1:6c|Solution d|Solution 2.1:6d}}
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===Övning 2.1:7===
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===Exercise 2.1:7===
<div class="ovning">
<div class="ovning">
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Bestäm koefficienterna framför <math>\,x\,</math> och <math>\,x^2\</math> när följande uttryck utvecklas
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Simplify the following by writing them as a single ordinary fraction
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{| width="100" cellspacing="10px"
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{| width="100%" cellspacing="10px"
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|a)
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|width="33%" | <math>\displaystyle \frac{2}{x+3}-\frac{2}{x+5}</math>
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|b)
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|width="33%" | <math>x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}</math>
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|c)
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|width="33%" | <math>\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:7|Solution a|Solution 2.1:7a|Solution b|Solution 2.1:7b|Solution c|Solution 2.1:7c}}
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===Exercise 2.1:8===
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<div class="ovning">
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Simplify the following fractions by writing them as a single ordinary
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{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33" | <math>\displaystyle \frac{2}{x+3}-\frac{2}{x+5}</math>
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|width="33%" | <math>\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }</math>
|b)
|b)
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|width="33%"| <math>x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}</math>
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|width="33%" | <math>\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}</math>
|c)
|c)
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|width="33%"| <math>\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}</math>
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|width="33%" | <math>\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 2.1:7|Lösning a|Lösning 2.1:7a|Lösning b|Lösning 2.1:7b|Lösning c|Lösning 2.1:7c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.1:8|Solution a|Solution 2.1:8a|Solution b|Solution 2.1:8b|Solution c|Solution 2.1:8c}}

Current revision

       Theory          Exercises      


Exercise 2.1:1

Expand

a) 3x(x−1) b) (1+x−x2)xy c) −x2(4−y2)
d) x3y2y1−1xy+1  e) (x−7)2 f) (5+4y)2
g) (y2−3x3)2 h) (5x3+3x5)2

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f | Solution g | Solution h


Exercise 2.1:2

Expand

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) (3x2+2)(3x2−2)(9x4+4)
e) (a+b)2+(a−b)2

Answer | Solution a | Solution b | Solution c | Solution d | Solution e

Exercise 2.1:3

Factorise and simplify as much as possible

a) x2−36 b) 5x2−20 c) x2+6x+9
d) x2−10x+25 e) 18x−2x3 f) 16x2+8x+1

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f

Exercise 2.1:4

Determine the coefficients in front of x and x2  when the following expressions are expanded out.

a) (x+2)(3x2−x+5)
b) (1+x+x2+x3)(2−x+x2+x4)
c) (x−x3+x5)(1+3x+5x2)(2−7x2−x4)

Answer | Solution a | Solution b | Solution c

Exercise 2.1:5

Simplify as much as possible

a) 1x−x2−x1 b) 1y2−2y−2y2−4
c) (x+1)(x+2)(3x2−12)(x2−1) d) (y2+4)(y2−4)(y2+4y+4)(2y−4)

Answer | Solution a | Solution b | Solution c | Solution d

Exercise 2.1:6

Simplify as much as possible

a) x−y+x2y−x  y2x−y−1  b) xx−2+xx+3−2
c) 2a+ba2−ab−2a−b d) \displaystyle \displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}

Answer | Solution a | Solution b | Solution c | Solution d

Exercise 2.1:7

Simplify the following by writing them as a single ordinary fraction

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

Answer | Solution a | Solution b | Solution c

Exercise 2.1:8

Simplify the following fractions by writing them as a single ordinary

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

Answer | Solution a | Solution b | Solution c

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