2.1 Exercises

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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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Skriv på gemensamt bråkstreck
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Expand
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 17: Line 17:
|width="33%" | <math>(1+x-x^2)xy</math>
|width="33%" | <math>(1+x-x^2)xy</math>
|c)
|c)
-
|width="33%" | <math>\displaystyle \frac{2}{7}-\frac{1}{5}</math>
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|width="33%" | <math>-x^2(4-y^2)</math>
|-
|-
|d)
|d)
-
|| <math>\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}</math>
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|| <math>x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right)</math>
|e)
|e)
-
||<math> \displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}</math>
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||<math> (x-7)^2</math>
|f)
|f)
-
||<math> \displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}</math>
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||<math> (5+4y)^2</math>
|-
|-
|g)
|g)
-
||<math> \displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}</math>
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||<math>(y^2-3x^3)^2</math>
|h)
|h)
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||<math> \displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}</math>
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||<math> (5x^3+3x^5)^2</math>
|}
|}
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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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 +
 
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===Exercise 2.1:2===
 +
<div class="ovning">
 +
Expand
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="50%"| <math>(x-4)(x-5)-3x(2x-3)</math>
 +
|b)
 +
|width="50%"| <math>(1-5x)(1+15x)-3(2-5x)(2+5x)</math>
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|-
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|c)
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|| <math>(3x+4)^2-(3x-2)(3x-8)</math>
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|d)
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|| <math>(3x^2+2)(3x^2-2)(9x^4+4)</math>
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|-
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|e)
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||<math> (a+b)^2+(a-b)^2</math>
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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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 +
===Exercise 2.1:3===
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<div class="ovning">
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Factorise and simplify as much as possible
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%" | <math>x^2-36</math>
 +
|b)
 +
|width="33%" | <math>5x^2-20</math>
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|c)
 +
|width="33%" | <math>x^2+6x+9</math>
 +
|-
 +
|d)
 +
|| <math>x^2-10x+25</math>
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|e)
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||<math> 18x-2x^3</math>
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|f)
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||<math> 16x^2+8x+1</math>
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|}
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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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===Exercise 2.1:4===
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<div class="ovning">
 +
Determine the coefficients in front of <math>\,x\,</math> and <math>\,x^2\</math> when the following expressions are expanded out.
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="100%" | <math>(x+2)(3x^2-x+5)</math>
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|-
 +
|b)
 +
|| <math>(1+x+x^2+x^3)(2-x+x^2+x^4)</math>
 +
|-
 +
|c)
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|| <math>(x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)</math>
 +
|-
 +
|}
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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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 +
===Exercise 2.1:5===
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<div class="ovning">
 +
Simplify as much as possible
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="50%" | <math>\displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x}</math>
 +
|b)
 +
|width="50%" | <math>\displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}</math>
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|-
 +
|c)
 +
|width="50%" | <math>\displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)}</math>
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|d)
 +
|| <math>\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}</math>
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|}
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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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===Exercise 2.1:6===
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<div class="ovning">
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Simplify as much as possible
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{| width="100%" cellspacing="10px"
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|a)
 +
|width="50%" | <math>\left(x-y+\displaystyle\frac{x^2}{y-x}\right)</math> <math>\left(\displaystyle\frac{y}{2x-y}-1\right)</math>
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|b)
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|width="50%" | <math>\displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2</math>
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|-
 +
|c)
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|width="50%" | <math>\displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b}</math>
 +
|d)
 +
|| <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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|}
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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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 +
===Exercise 2.1:7===
 +
<div class="ovning">
 +
Simplify the following by writing them as a single ordinary fraction
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%" | <math>\displaystyle \frac{2}{x+3}-\frac{2}{x+5}</math>
 +
|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)
 +
|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)
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|width="33%" | <math>\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }</math>
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|b)
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|width="33%" | <math>\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}</math>
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|c)
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|width="33%" | <math>\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}</math>
 +
|}
 +
</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) \displaystyle 3x(x-1) b) \displaystyle (1+x-x^2)xy c) \displaystyle -x^2(4-y^2)
d) \displaystyle x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right) e) \displaystyle (x-7)^2 f) \displaystyle (5+4y)^2
g) \displaystyle (y^2-3x^3)^2 h) \displaystyle (5x^3+3x^5)^2
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


Exercise 2.1:2

Expand

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

Exercise 2.1:3

Factorise and simplify as much as possible

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

Exercise 2.1:4

Determine the coefficients in front of \displaystyle \,x\, and \displaystyle \,x^2\ when the following expressions are expanded out.

a) \displaystyle (x+2)(3x^2-x+5)
b) \displaystyle (1+x+x^2+x^3)(2-x+x^2+x^4)
c) \displaystyle (x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)
Answer
Answer 2.1:4
Solution a
Solution 2.1:4a
Solution b
Solution 2.1:4b
Solution c
Solution 2.1:4c

Exercise 2.1:5

Simplify as much as possible

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

Exercise 2.1:6

Simplify as much as possible

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

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
Answer 2.1:7
Solution a
Solution 2.1:7a
Solution b
Solution 2.1:7b
Solution c
Solution 2.1:7c

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
Answer 2.1:8
Solution a
Solution 2.1:8a
Solution b
Solution 2.1:8b
Solution c
Solution 2.1:8c
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