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

Answer with explanation:

The meaning of perfect square for a polynomial can be written as ,if a polynomial having one variable can be expressed as

$(ax^n+b)^2$

We have to find the value of ,a,b,and c such that ,$x^4+ax^3+bx^2+cx+1 {\text{and}} x^4+2ax^3+2bx^2+2cx+1$ becomes a perfect square.

$1.x^4+ax^3+bx^2+cx+1$

put,a=0,b=2,c=0

The above equation becomes

$x^4+2x^2+1=x^2+2\times x^2+1=(x^2+1)^2$

$2.x^4+2ax^3+2bx^2+2cx+1$

Put,a=0,b=1,c=0

The above equation becomes

$x^4+2x^2+1\\\\=(x^2)^2+2\times x^2+1^2\\\\=(x^2+1)^2$

Used the following Identity

$(a+b)^2=a^2+b^2+2ab$

For, Polynomial 1, →a=0,b=2,c=0

For,Polynomial ,2,→a=0,b=1,c=0

If we substitute values of ,a,b and c in the above two Polynomial, it will become a perfect square.