Question

Simplify the expression (2x^2 + 5x - 3) / (x + 2) using only positive exponents.

Answer 1

To simplify the expression \((2x^2 + 5x - 3) / (x + 2)\), we can perform polynomial long division.

1. **Divide the leading term**:
Divide the leading term of the numerator \(2x^2\) by the leading term of the denominator \(x\):
\[
2x^2 \div x = 2x
\]

2. **Multiply and subtract**:
Multiply \(2x\) by the entire denominator \(x + 2\):
\[
2x(x + 2) = 2x^2 + 4x
\]
Now, subtract this from the original polynomial:
\[
(2x^2 + 5x - 3) - (2x^2 + 4x) = (5x - 4x) - 3 = x - 3
\]

3. **Repeat the process**:
Now divide the leading term of \(x - 3\) by the leading term of \(x + 2\):
\[
x \div x = 1
\]
Multiply \(1\) by the entire denominator:
\[
1(x + 2) = x + 2
\]
Subtract:
\[
(x - 3) - (x + 2) = -3 - 2 = -5
\]

At this point, we can't divide further because the degree of \(-5\) (which is 0) is less than the degree of \(x + 2\) (which is 1).

So, the result of our long division can be expressed as:
\[
2x + 1 - \frac{5}{x + 2}
\]

Therefore, the simplified expression is:
\[
2x + 1 - \frac{5}{x + 2}
\]

You can write this final result as:
\[
2x + 1 - 5(x + 2)^{-1}
\]

This is the simplified expression using only positive exponents.