Breath Math

1. Factorise:

(i) 8y³ – 1

Solution:

= (2y)³ – 1³ [∵ a³ – b³ = (a – b) (a² + ab + b²)]

= (2y – 1) [(2y)² + 2y .1 + 1²]

= (2y – 1) (4y² + 2y + 1)

 

(ii) 27x³ – 8

Solution:

= (3x)³ – 2³ [∵ a³ – b³ = (a – b) (a² + ab + b²)]

= (3x – 2) [(3x)² + 3x .2 + 2²]

= (3x – 2) (9x² + 6x + 4)

 

(iii) x³ + 8y³

Solution:

= x³ + (2y)³ [∵ a³ + b³ = (a + b) (a² – ab + b²)]

= (x + 2y) [x² – x.2y + (2y)²]

= (x + 2y) (x² – 2xy + 4y)²

 

(iv) 1 – x³

Solution:

= 1³ – x³ [∵a³ – b³ = (a – b) (a² + ab + b²)]

= (1 – x) (1² + 1.x + x²)

= (1 – x) (1 + x + x²)

 

(v) a³ b³ + c³

Solution:

= (ab)³ + c³ [∵ a³ + b³ = (a + b) (a² – ab + b²)]

= (ab + c) [(ab)² – ab .c + c²]

= (ab + c) (a²b² – abc + c²)

 

(vi) a³b – ^𝐛/_𝟔𝟒

Solution:

= b (a³– ¹/₆₄)

= b [a³ – (¹/₄ )³]

= b (a – ¹/₄) [a² + a. ¹/₄ + ( ¹/₄ )²]

= b (a – 14) (a² + ^a/₄ + ¹/₁₆ )

 

(vii) 𝐚³/_𝟖 + 1

Solution:

= ( ^a/₂ )³ + 1³ [∵ a³ + b³ = (a + b) (a² – ab + b²)]

= ( ^a/₂ + 1) [( ^a/₂ )² – ^a/₂ .1 + 1²]

= ( ^a/₂ + 1) [a²/₄− ^a/₃+ 1]

 

(viii) 3a⁶ – 𝐛^𝟔/_𝟗

Solution:

= 3(a⁶− b⁶/27)

= 3[(a²)³− (b²/₃)³]

= 3[(a²− b²/₃) (a²)³− a².b²/₃ (b²/₃)²

= 3(a²− b²/₃)( a⁴+a² . b²/₃+ b⁴/₃ )

 

(ix) 2a³ + ^𝟏/_𝟒

Solution:

= 2 (a³ + ^𝟏/_𝟒 )

= 2 (a³ + ^𝟏/₂ )³

= 2 (a + ^𝟏/₂ ) [ a² – a. ^𝟏/₂ + (^𝟏/_𝟒 )²]

= 2 (a + ^𝟏/₂ ) (a² – ^a/₂ + ^𝟏/_𝟒 )

 

(x) x³ – 512

Solution:

= x³ – 8³ [∵ a³ – b³ = (a – b) (a² + ab + b²)]

= (x – 8) (x² + 8x + 64)

 

(xi) 32x³ – 500

= 4 (8x³ – 125)

= 4 [(2x)³ – 5³]

= 4 (2x – 5) [(2x)² + 2x .5 – 5²]

= 4 (2x – 5) (4x² + 10 x + 25)

 

(xii) x⁷ + xy⁶

= x (x⁶ + y⁶)

= x [(x²)³ + (y²)³]

= x [(x² + y²) { (x²)² – x² y² + (y²)²}]

= x (x² + y²) (x⁴ – x² y² + y²)

 

(xiii) 2a⁴ – 128a

= 2a (a³ – 64)

= 2a (a³ – 4³)

= 2a (a – 4) (a² + 4a + 4²)

= 2a (a – 4) (a² + 4a + 16)

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2. Factorise

(i) (1 – a)³ + (3a)³

Solution:

Using x³ + y³ = (x + y) (x² – xy + y²)

(1 – a)³ + (3a)³

= (1 – a + 3a) [(1 – a)² – (1 – a) 3a + (3a)²]

= (1 + 2a) [(1 – a)² – 3a + 3a² + 9a²]

= (1 + 2a) (1 + a² – 2a – 3a + 12a²)

= (1 + 2a) (13a² – 5a + 1)

 

(ii) 8x³ – 27y³

Solution:

= (2x)³ – ( 3y)³

= ( 2x – 3y) [(2x)² + 2x .3y + (3y)²]

= (2x – 3y) (4x² + 6xy + 9y²)

 

(iii) z⁴ x³ + 8y³ z⁴

Solution:

= z⁴ (x³ + 8y³)

= z⁴ [x³ + (2y)³]

= z⁴ (x + 2y) [x² – x.2y + (2y)²]

= z⁴ (x + 2y) [x² – 2xy + 4y²]

 

(iv) 3(x + y)³ + ^𝟏/₉ (xy)³

Solution:

=  (x + y)³ + ¹/₂₇ (xy)³

= (x + y)³ + ( ^xy/₃ )³

= (x + y + ^xy/₃ ) (x + y)² – [ (x + y) ( ^xy/₃ )+ ( ^xy/₃)²]

= (x + y + ^xy/₃ ) (x² + y²  + 2xy – (x + y)( ^xy/₃) + x²y²/₉ )

 

(v) x⁶ + y⁶

Solution:

= (x²)³ – (y²)³

= (x² – y²) [(x²)³ + x² y²– (y²)³]

= (x² – y²) (x⁴ + x² y² + y⁴)

= (x + y) (x – y) (x² + x y + y²) (x² – xy + y²)

 

(vi) a³ – 2√𝟐 b³

Solution:

= a³ – (√2 b)³

= (a –√2b) [a² + a.√2 b + √2 b²]

= (a –√𝟐 b) (a² + √𝟐 ab + √𝟐b²)

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3. Factorize the following

(i) x⁶ – 26x³ – 27

Solution:

put x³ = a

a³ – 26a – 27

= a³ – 27a + a – 27

= a ( a – 27) + 1 (a – 27)

= (a – 27) ( a + 1)

= (x³ – 27) (x³ + 1)

= (x – 3) (x² + 3x + 9) (x – 1) (x² – x + 1)

 

(ii) z⁶ – 63z³ – 64

Solution:

put z³ = a

a³ – 63a – 64

= a³ – 64a + a – 64

= a (a – 64) + 1 (a – 64)

= (a – 64) (a + 1)

= (z³ – 64) (z³ + 1)

= (z – 4) (z² + 4z + 16) (z + 1) (z² – z + 1)

 

(iii) a³ – b³ – a + b

Solution:

= (a – b) (a² + ab + b²) – (a – b)

= (a – b) [ a² + ab + b² – 1]

 

(iv) x⁶ + 7x³ – 8

Solution:

put x³ = a

a³ + 7a – 8

= a³ + 8a – a – 8

= a (a + 8) – 1 (a + 8)

= (a – 1) (a + 8)

= (x³ – 1) (x³ + 8)

= (x – 1) (x² + x – 1) (x + 2) (x² – 2x + 4)

 

(v) a³ – ^𝟏/_𝐚^𝟑 – 2a + 𝟐𝐚

Solution:

= (a – ^𝟏/_𝐚) (a² + a. ^𝟏/_𝐚+ ^𝟏/_𝐚² ) – 2 (a – ^𝟏/_𝐚 )

= (a – ^𝟏/_𝐚 ) [a² + 1 + ^𝟏/_𝐚² – 2]

= (a – ^𝟏/_𝐚 ) (a² + ^𝟏/_𝐚^𝟐 – 1 )

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