introduction in this chapter you will learn to add fractions with different denominators (a recap)...
TRANSCRIPT
![Page 1: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/1.jpg)
Partial Fractions
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Introduction
β’ In this chapter you will learn to add fractions with different denominators (a recap)
β’ You will learn to work backwards and split an algebraic fraction into components called βPartial Fractionsβ
![Page 3: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/3.jpg)
Teachings for Exercise 1A
![Page 4: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/4.jpg)
Partial FractionsYou can add and subtract
several fractions as long as they share a common
denominator
You will have seen this plenty of times already! If you want to
combine fractions you must make the denominators equivalentβ¦
1A
Calculate:1338+ΒΏ
824
924+ΒΏ
1724ΒΏ
Γ88
Γ33
![Page 5: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/5.jpg)
Partial FractionsYou can add and subtract
several fractions as long as they share a common
denominator
You will have seen this plenty of times already! If you want to
combine fractions you must make the denominators equivalentβ¦
1A
Calculate: 2π₯+3
1π₯+1β
2(π₯+1)(π₯+3)(π₯+1)
1(π₯+3)(π₯+3)(π₯+1)β
Γπ₯+1π₯+1
Γπ₯+3π₯+3
2π₯+2(π₯+3)(π₯+1)
π₯+3(π₯+3)(π₯+1)β
π₯β1(π₯+3)(π₯+1)ΒΏ
Multiply brackets
Group terms
![Page 6: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/6.jpg)
Teachings for Exercise 1B
![Page 7: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/7.jpg)
Partial FractionsYou can split a fraction with
two linear factors into Partial Fractions
1B
For example:π₯β1
(π₯+3)(π₯+1)2
π₯+31
π₯+1βΒΏ when split up into Partial Fractions
11(π₯β3)(π₯+2)
π΄π₯β3
π΅π₯+2+ΒΏΒΏ when split up into Partial Fractions
You need to be able to calculate the values of A and Bβ¦
![Page 8: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/8.jpg)
Partial FractionsYou can split a fraction with
two linear factors into Partial Fractions
1B
Split
6 π₯β2(π₯β3)(π₯+1)into Partial Fractions
6 π₯β2(π₯β3)(π₯+1)
π΄(π₯β3)
π΅(π₯+1)+ΒΏ
π΄(π₯+1)(π₯β3)(π₯+1)
π΅(π₯β3)(π₯β3)(π₯+1)+ΒΏ
ΒΏπ΄ (π₯+1 )+π΅(π₯β3)
(π₯β3)(π₯+1)
6 π₯β2ΒΏA (π₯+1 )+π΅(π₯β3)β8ΒΏβ4π΅2ΒΏπ΅16ΒΏ4 π΄4ΒΏπ΄
ΒΏ4
(π₯β3)2
(π₯+1)+ΒΏ
Split the Fraction into its 2 linear parts, with numerators A and B
Cross-multiply to make the denominators the same
Group together as one fraction
This has the same denominator as the initial fraction, so the
numerators must be the same
If x = -1:
If x = 3:
You now have the values of A and B and can write the answer
as Partial Fractions
![Page 9: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/9.jpg)
Teachings for Exercise 1C
![Page 10: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/10.jpg)
Partial FractionsYou can also split fractions
with more than 2 linear factors in the denominator
1C
For example:4
(π₯+1 ) (π₯β3 )(π₯+4)π΄
π₯+1π΅
π₯β3+ΒΏΒΏπΆπ₯+4+ΒΏ
when split up into Partial Fractions
![Page 11: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/11.jpg)
Partial FractionsYou can also split fractions
with more than 2 linear factors in the denominator
1C
Split
into Partial fractions
6 π₯2+5π₯β2π₯ (π₯β1)(2π₯+1)
6 π₯2+5π₯β2π₯ (π₯β1)(2π₯+1)
π΄π₯
π΅π₯β1
πΆ2π₯+1+ΒΏ +ΒΏ
π΄(π₯β1)(2π₯+1)π₯ (π₯β1)(2 π₯+1)
π΅(π₯ )(2π₯+1)π₯ (π₯β1)(2π₯+1)+ΒΏ πΆ (π₯)(π₯β1)
π₯ (π₯β1)(2π₯+1)+ΒΏπ΄ (π₯β1 ) (2 π₯+1 )+π΅ (π₯ ) (2 π₯+1 )+πΆ (π₯)(π₯β1)
π₯ (π₯β1)(2π₯+1)
π΄ (π₯β1 ) (2 π₯+1 )+π΅ (π₯ ) (2π₯+1 )+πΆ (π₯)(π₯β1)6 π₯2+5π₯β2ΒΏ9ΒΏ 3
3ΒΏπ΅β2ΒΏβπ΄2ΒΏπ΄β3ΒΏ0.75πΆβ4ΒΏπΆ
2π₯
3π₯β1
42π₯+1+ΒΏ βΒΏ
Split the Fraction into its 3 linear parts
Cross Multiply to make the
denominators equal
Put the fractions together
The numerators must be equal
If x = 1
If x = 0
If x = -0.5
You can now fill in the
numerators
![Page 12: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/12.jpg)
Partial FractionsYou can also split fractions
with more than 2 linear factors in the denominator
1C
Split
into Partial fractions
4 π₯2β21π₯+11π₯3β4 π₯2+π₯+6
You will need to factorise the
denominator firstβ¦
π₯3β4 π₯2+π₯+6(1)3β4 (1 )2+(1)+6ΒΏ 4(β1)3β4 (β1 )2+(β1)+6 0
Therefore (x + 1) is a factorβ¦
π₯3β4 π₯2+π₯+6π₯+1π₯3+π₯2
β5 π₯2+π₯+6
π₯2β5 π₯
β5 π₯2β5π₯6 π₯+6
+6
6 π₯+60
Try substituting factors to make the expression 0
Divide the expression by (x + 1)
π₯3β4 π₯2+π₯+6ΒΏ (π₯+1)(π₯2β5 π₯+6)
π₯3β4 π₯2+π₯+6 (π₯+1)(π₯β2)(π₯β3)ΒΏYou can now factorise
the quadratic part
![Page 13: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/13.jpg)
Partial FractionsYou can also split fractions
with more than 2 linear factors in the denominator
1C
Split
into Partial fractions
4 π₯2β21π₯+11π₯3β4 π₯2+π₯+6
4 π₯2β21π₯+11π₯3β4 π₯2+π₯+6
ΒΏ 4 π₯2β21π₯+11(π₯+1)(π₯β2)(π₯β3)
π΄π₯+1
π΅π₯β2+ΒΏ πΆ
π₯β3+ΒΏπ΄(π₯β2)(π₯β3)
(π₯+1)(π₯β2)(π₯β3)π΅(π₯+1)(π₯β3)
(π₯+1)(π₯β2)(π₯β3)+ΒΏ+ΒΏ πΆ (π₯+1)(π₯β2)(π₯+1)(π₯β2)(π₯β3)
π΄ (π₯β2 ) (π₯β3 )+π΅ (π₯+1 ) (π₯β3 )+πΆ (π₯+1)(π₯β2)(π₯+1)(π₯β2)(π₯β3)
π΄ (π₯β2 ) (π₯β3 )+π΅ (π₯+1 ) (π₯β3 )+πΆ (π₯+1)(π₯β2)4 π₯2β21 π₯+11β15ΒΏΒΏβ3 π΅5ΒΏπ΅16ΒΏ C
β4ΒΏπΆ
If x = 2
If x = 3
If x = -1 36ΒΏ A
3ΒΏπ΄3
π₯+15
π₯β2+ΒΏ 4π₯β3βΒΏ
Split the fraction into its 3 linear parts
Cross multiply
Group the fractions
Replace A, B and
C
The numerators must be
equal
![Page 14: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/14.jpg)
Teachings for Exercise 1D
![Page 15: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/15.jpg)
Partial FractionsYou need to be able to split a fraction that has repeated
linear roots into a Partial Fraction
1D
For example:3π₯2β4 π₯+2
(π₯+1 ) ΒΏΒΏπ΄
(π₯+1)π΅
(π₯β5)+ΒΏΒΏ πΆΒΏΒΏ+ΒΏ when split up into
Partial Fractions
The repeated root is included once βfullyβ
and once βbroken downβ
![Page 16: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/16.jpg)
Partial FractionsYou need to be able to split a fraction that has repeated
linear roots into a Partial Fraction
1D
Split
into Partial fractions
11π₯2+14 π₯+5ΒΏΒΏ
11π₯2+14 π₯+5ΒΏΒΏ
π΄(π₯+1)
π΅ΒΏΒΏ
πΆ(2 π₯+1)+ΒΏ +ΒΏ
π΄(π₯+1)(2 π₯+1)ΒΏΒΏ
+ΒΏ+ΒΏπ΅(2 π₯+1)ΒΏΒΏ πΆ ΒΏΒΏ
ΒΏ π΄ (π₯+1 ) (2π₯+1 )+π΅ (2π₯+1 )+πΆΒΏΒΏπ΄ (π₯+1 ) (2π₯+1 )+π΅ (2π₯+1 )+πΆ ΒΏΒΏ11π₯2+14 π₯+52ΒΏβπ΅β2ΒΏπ΅0.75ΒΏ3ΒΏπΆ
If x = -1
If x = -0.5 0.25πΆAt this point there is no way to cancel B and C to leave A
by substituting a value in
Choose any value for x (that hasnβt been used yet), and
use the values you know for B and C to leave A
If x = 0 ΒΏ5 1 π΄+ΒΏ1π΅+ΒΏ C
ΒΏ5 π΄β2+ΒΏ3
ΒΏ4 π΄4
(π₯+1)2ΒΏΒΏ
3(2 π₯+1)
β +ΒΏΒΏ
Split the fraction into its 3 parts
Make the denominators
equivalent
Group up
The numerators will be the same
Sub in the values of A, B and C
![Page 17: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/17.jpg)
Teachings for Exercise 1E
![Page 18: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/18.jpg)
Partial FractionsYou can split an improper fraction
into Partial Fractions. You will need to divide the numerator by the denominator first to find the
βwholeβ part
1E
2235ΒΏ
1537+ΒΏ
5720ΒΏ
1435+ΒΏ2+ΒΏ
A regular fraction being split into 2
βcomponentsβ
A top heavy (improper) fraction will have a βwhole
number part before the fractions
![Page 19: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/19.jpg)
Partial FractionsYou can split an improper fraction
into Partial Fractions. You will need to divide the numerator by the denominator first to find the
βwholeβ part
1E
Split
into Partial fractions
3 π₯2β3π₯β2(π₯β1)(π₯β2)
Remember, Algebraically an βimproperβ fraction is one where
the degree (power) of the numerator is equal to or
exceeds that of the denominator
3 π₯2β3π₯β2(π₯β1)(π₯β2)
3π₯2β3 π₯β2π₯2β3 π₯+2ΒΏ
π₯2β3 π₯+23 π₯2β3π₯β23
3 π₯2β9π₯+66 π₯β8
3 π₯2β3π₯β2(π₯β1)(π₯β2)ΒΏ
6 π₯β8(π₯β1)(π₯β2)
3+ΒΏ
π΄(π₯β1)
π΅(π₯β2)+ΒΏ
ΒΏπ΄ (π₯β2 )+π΅(π₯β1)
(π₯β1)(π₯β2)
6 π₯β8ΒΏπ΄ (π₯β2 )+π΅(π₯β1)If x = 2 4ΒΏπ΅If x = 1 β2ΒΏβπ΄
2ΒΏπ΄
Divide the numerator by the denominator to find the βwholeβ part
Now rewrite the original fraction with the whole
part taken out
Split the fraction into 2 parts (ignore the whole
part for now)
Make denominators equivalent and group
up
The numerators will be the same
ΒΏ3+ΒΏ 2(π₯β1)
4(π₯β2)+ΒΏ
![Page 20: Introduction In this chapter you will learn to add fractions with different denominators (a recap) You will learn to work backwards and split an algebraic](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649dce5503460f94ac2a10/html5/thumbnails/20.jpg)
Summary
β’ We have learnt how to split Algebraic Fractions into βPartial fractionsβ
β’ We have also seen how to do this when there are more than 2 components, when one is repeated and when the fraction is βimproperβ