Question

The expression x^3+2x^2+x+6/x+2 is equivalent to

A) x^2+3
B)x^2+1+4/x+2
C) 2x^2+ x+6
D) 2x^2+1+4/x+2

Answer 1

The expression x^3+2x^2+x+6/x+2 simplifies to x^2 + 4/(x + 2) after performing polynomial long division, which matches choice B: x^2 + 1 + 4/(x + 2).

Step-by-step explanation:

The expression x^3+2x^2+x+6/x+2 is being asked to simplify. To find the equivalent expression, we can use polynomial long division or synthetic division to divide the numerator by the denominator. Let's perform polynomial long division:

1. Divide the first term in the numerator by the first term in the denominator: x^3 / x = x^2.
2. Multiply the entire divisor (x + 2) by this result (x^2) and subtract it from the numerator.
3. The result is the new numerator. Repeat the process using this new numerator.
4. Continue this process until the degree of the remainder is less than the degree of the divisor.

After performing this division, the result is x^2 with a remainder of 4. Therefore, the expression simplifies to x^2 + 4/(x + 2). This matches choice B: x^2 + 1 + 4/(x + 2).