Question

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

Answer 1

Answer:

Option C

Step-by-step explanation:

Given expression is $\frac{-5x^3+2x^2+9x+2}{(x^2+2)^2}$

Option A

$\frac{A}{x^2+2}+ \frac{Bx+C}{(x^2+2)^2}$

= $\frac{A(x^2+2)}{(x^2+2)^2}+ \frac{Bx+C}{(x^2+2)^2}$

= $\frac{Ax^2+2A+Bx+C}{(x^2+2)^2}$

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

Therefore, both the expressions are not equivalent.

Option B

$\frac{A}{(x^2+2)^2}+ \frac{Bx+C}{(x^2+2)^2}$

= $\frac{A+Bx+C}{(x^2+2)^2}$

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

$\frac{Ax+B}{(x^2+2)}+ \frac{Cx+D}{(x^2+2)^2}$

= $\frac{(Ax+B)(x^2+2)}{(x^2+2)^2}+ \frac{Cx+D}{(x^2+2)^2}$

= $\frac{Ax^3+Bx^2+2Ax+2B+Cx+D}{(x^2+2)^2}$

= $\frac{Ax^3+Bx^2+x(2A+C)+D}{(x^2+2)^2}$

Here, numerator is a cubic polynomial.

Therefore, both the expressions are equivalent.

Option D

$\frac{Ax+B}{(x^2+2)^2}+ \frac{Cx+D}{(x^2+2)^2}$

= $\frac{x(A+C)+(B+D)}{(x^2+2)^2}$

Numerator is a linear polynomial.

Therefore, sum is not equal to the given expression.

Option C will be the answer.