Question

Find out the value of a,b,c which will make each of expression x^4+ax^3+bx^2+cx+1 and x^4+2ax^3+2bx^2+2cx+1 a perfect square

Answer 1

given expression , x⁴ + ax³ + bx² + cx + 1

in this expression it is clearly shown that coefficient of x⁴ and constant term are 1.then square root of it must be contain coefficient of x² and constant are 1

that means this expression will be in the form of (x² + kx + 1)², where k is real numbers.

or, x⁴ + ax³ + bx² + cx + 1 = (x² + kx + 1)²

= x⁴ + k²x² + 1² + 2kx³ + 2kx + 2x²

= x⁴ + 2kx³ + (k² + 2)x² + 2kx + 1

comparing both sides,

a = 2k, b = k² + 2 , c = 2k

if we assume k = 1

then, a = 2, b = 3 and c = 2 [ Ans]
you should try to put a, b, c in second expression.
x⁴ + 2 × 2x³ + 2 × 3x² + 2 × 2x + 1
= x⁴ + 4x³ + 6x² + 4x + 1
= x⁴ + 2x³ + x² + 2x³ + 4x² + 2x + x² + 2x + 1
= x²(x² + 2x + 1) + 2x(x² + 2x + 1) + 1(x² + 2x + 1)
= (x² + 2x + 1)²

hence, it is true that
a = 2, b = 3 and c = 2