The value of n for which the expression x4 + 4x3 + nx2 + 4x + 1 becom

1.	The value of n for which the expression x4 + 4x3 + nx2 + 4x + 1 becomes a perfect square is: 1. 3 2. 4 3. 5 4. 6
Answer» 4. 6 x⁴ + 4x³ + nx² + 4x + 1= (ax² + bx + c)² x⁴+ 4x³+ nx² + 4x + 1 = a² x⁴ + b² x² + c² +2abx³ + 2bcx + 2acx² = a² x⁴ + 2abx³ + (b² + 2ac ) x² + 2bcx + c² Comparing coefficients, a² = 1 c² = 1 2ab = 4 b2 + 2 ac = n 2bc = 4 Solving, we get a/c = 1 ⇒ a = c b = \(\pm\)2 , a = \(\pm\)1 , c = \(\pm\)1 ∴ b²+ 2ac = 4 + 2 = 6

The value of n for which the expression x4 + 4x3 + nx2 + 4x + 1 becomes a perfect square is: 1. 3 2. 4 3. 5 4. 6

Answer»

4. 6

x⁴ + 4x³ + nx² + 4x + 1= (ax² + bx + c)²

x⁴+ 4x³+ nx² + 4x + 1 = a² x⁴ + b² x² + c²

+2abx³ + 2bcx + 2acx²

= a² x⁴ + 2abx³ + (b² + 2ac ) x² + 2bcx + c²

Comparing coefficients,

a² = 1

c² = 1

2ab = 4

b2 + 2 ac = n

2bc = 4

Solving, we get a/c = 1 ⇒ a = c

b = \(\pm\)2 , a = \(\pm\)1 , c = \(\pm\)1

∴ b²+ 2ac = 4 + 2 = 6

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The value of n for which the expression x4 + 4x3 + nx2 + 4x + 1 becomes a perfect square is: 1. 3 2. 4 3. 5 4. 6

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