Algebraic Expressions and Identities Test - 6

(6x³ − y + 3x²y) × (x² + y²)

= x²(6x³ + 3x²y − y) + y²(6x³ + 3x²y − y)

= 6x⁵ + 3x⁴y − x²y + 6x³y² + 3x²y³ − y³

We have, 
3x + 4y = 18.................?(i) 

xy = 6.............................. ?(ii)
Squaring both sides of (i), we get 

(3x)² + (4y)² + 2 × 3x × 4y = (18)²

= 9x² + 16y² + 24xy = 324 

9x² + 16y² + 24(6) = 324........................(Using (ii)) 

9x² + 16y² + 144 = 324

⇒9x² + 16y² = 1

(9x−7xy)² = (9x)² − 2 × 9x × 7xy + (7xy)²

=81x² − 126x²y + 49x²y²

(x−1x)2 = x² + 1/x² − = 53 − 2 = 51

 ⇒ x − 1/x = √51.

We have, 

3x − 7y = 10      ...(i)

 And 

xy = −1          ...(ii)

Squaring both sides of (i), we get 

(3x)² + (7y)² − 2 × 3x × 7y = 10²

or 

9x² + 49y² − 42xy = 100

or 

9x² + 49y² − 42(−1) = 100

       (Using (i)) 

∴ 9x² + 49y² = 100 − 42 = 58

(x² + 3x + 5)(x² − 1)

= x4 − x² + 3x³ − 3x + 5x² − 5

 Grouping like terms, we get, 

x⁴ + 3x³ − x² + 5x² − 3x − 5

= x⁴ + 3x³ + 4x² − 3x − 5.

Let the expression to be added is x. 

⇒ 4p² + 5p + 7 + x = 7p² + 2p + 9

⇒ x = 7p² + 2p + 9 − 4p² − 5p − 7

⇒ x = 3p² − 3p + 2

 is the required expression.