## Algebra

#### Algebra

1. If for two real constants a and b, the expression ax3 + 3x2 - 8x + b is exactly divisible by (x + 2) and (x – 2), then

1. P (x ) = ax3 + 3x2 – 8x + b
[∵ P (x) is div. by (x + 2) & (x –2)]
∴ P (–2) = –8a + 12 + 16 + b = 0
⇒ –8a + b + 28 = 0 ...(i)
⇒ P(2) = 8a + 12 – 16 + b = 2
⇒ 8a + b – 4 = 0 ...(ii)
By equation (i) + (ii)
2b + 24 = 0

 ⇒ b = - 24 = -12 2

From equation (i),
– 8a – 12 + 28 = 0
⇒ –8a = –16 ⇒ a = 2

##### Correct Option: C

P (x ) = ax3 + 3x2 – 8x + b
[∵ P (x) is div. by (x + 2) & (x –2)]
∴ P (–2) = –8a + 12 + 16 + b = 0
⇒ –8a + b + 28 = 0 ...(i)
⇒ P(2) = 8a + 12 – 16 + b = 2
⇒ 8a + b – 4 = 0 ...(ii)
By equation (i) + (ii)
2b + 24 = 0

 ⇒ b = - 24 = -12 2

From equation (i),
– 8a – 12 + 28 = 0
⇒ –8a = –16 ⇒ a = 2

1.  If x + 1 = 3 , find the value of 8x3 + 1 4x 2 8x3

1. Using Rule 8,

 x + 1 = 3 4x 2

Multiplying both sides by 2
 ⇒ 2x + 1 = 3 2x

On cubing both sides,
 ∴ 8x3 + 1 + 3 × 2x × 1 × 2x + 1 = 27 8x3 2x 2x

 ⇒ 8x3 + 1 + 3 × 3 = 27 8x3

 ⇒ 8x3 + 1 = 27 – 9 = 18 8x3

##### Correct Option: A

Using Rule 8,

 x + 1 = 3 4x 2

Multiplying both sides by 2
 ⇒ 2x + 1 = 3 2x

On cubing both sides,
 ∴ 8x3 + 1 + 3 × 2x × 1 × 2x + 1 = 27 8x3 2x 2x

 ⇒ 8x3 + 1 + 3 × 3 = 27 8x3

 ⇒ 8x3 + 1 = 27 – 9 = 18 8x3

1.  If x2 - 3x - 8x + 1 = 0, then the value of x3 + 1 is x3

1. Using Rule 8,
x2 – 3x + 1 = 0
⇒ x2 + 1 = 3x

 ⇒ x + 1 = 3 x

 ∴ x3 + 1 = x + 1 3 - 3 × x × 1 x + 1 x3 x x x

= 27 – 3 × 3 = 18

##### Correct Option: B

Using Rule 8,
x2 – 3x + 1 = 0
⇒ x2 + 1 = 3x

 ⇒ x + 1 = 3 x

 ∴ x3 + 1 = x + 1 3 - 3 × x × 1 x + 1 x3 x x x

= 27 – 3 × 3 = 18

1.  If 1 = 1 + 1 ( x ≠ 0,x ≠ 0 , x ≠ y ) then, the value of x3 - y3 is x + y x y

1.  1 = 1 + 1 = y + x x + y x y xy

⇒ ( x + y )2 = xy
⇒ x2 + 2xy + y2 = xy
⇒ x2 + 2xy + y2 - xy = 0
⇒ x2 + xy + y2 = 0
∴ x3 - y3 = (x – y)( x2 + xy + y2 ) = 0

##### Correct Option: A

 1 = 1 + 1 = y + x x + y x y xy

⇒ ( x + y )2 = xy
⇒ x2 + 2xy + y2 = xy
⇒ x2 + 2xy + y2 - xy = 0
⇒ x2 + xy + y2 = 0
∴ x3 - y3 = (x – y)( x2 + xy + y2 ) = 0

1.  If x + 1 = 2 ,find the value of 8x3 + 1 . 2x x3

1. Using Rule 8,

 x + 1 = 2 2x

 ⇒ 2x + 2 = 2 × 2 = 4 2x

 ⇒ 2x + 1 = 4 x

On cubing both sides,
 ∴ 8x3 + 1 + 3 × 2x × 1 2x + 1 = 64 x3 x x

 ⇒ 8x3 + 1 + 6 × 4 = 64 x3

 ⇒ 8x3 + 1 = 64 – 24 = 40 x3

##### Correct Option: C

Using Rule 8,

 x + 1 = 2 2x

 ⇒ 2x + 2 = 2 × 2 = 4 2x

 ⇒ 2x + 1 = 4 x

On cubing both sides,
 ∴ 8x3 + 1 + 3 × 2x × 1 2x + 1 = 64 x3 x x

 ⇒ 8x3 + 1 + 6 × 4 = 64 x3

 ⇒ 8x3 + 1 = 64 – 24 = 40 x3