Question

Which sum is equal to the expression - 5x3 + 2x2 + 9x+2/
(x²+ 2)²

Answer 1

Option C

Explanation:

Given expression is
`(-5x^3+2x^2+9x+2)/((x^2+2)^2)`

Option A

`(A)/(x^2+2)+ (Bx+C)/((x^2+2)^2)`

=
`(A(x^2+2))/((x^2+2)^2)+ (Bx+C)/((x^2+2)^2)`

=
`(Ax^2+2A+Bx+C)/((x^2+2)^2)`

Numerator is a quadratic polynomial, while the expression has the numerator as a cubic .

Therefore, both the expressions are not equivalent.

Option B

`(A)/((x^2+2)^2)+ (Bx+C)/((x^2+2)^2)`

=
`(A+Bx+C)/((x^2+2)^2)`

Here, numerator of the expression is a linear polynomial which is not equal to given expression.

Therefore, both the expressions are not equivalent.

Option C

`(Ax+B)/((x^2+2))+ (Cx+D)/((x^2+2)^2)`

=
`((Ax+B)(x^2+2))/((x^2+2)^2)+ (Cx+D)/((x^2+2)^2)`

=
`(Ax^3+Bx^2+2Ax+2B+Cx+D)/((x^2+2)^2)`

=
`(Ax^3+Bx^2+x(2A+C)+D)/((x^2+2)^2)`

Here, numerator is a cubic polynomial.

Therefore, both the expressions are equivalent.

Option D

`(Ax+B)/((x^2+2)^2)+ (Cx+D)/((x^2+2)^2)`

=
`(x(A+C)+(B+D))/((x^2+2)^2)`

Numerator is a linear polynomial.

Therefore, sum is not equal to the given expression.

Option C will be the answer.