Question
How do I simplify the expression (x^2 + 5x + 6) / (x + 2)
Answers
GPT-4o mini
To simplify the expression \((x^2 + 5x + 6) / (x + 2)\), you can start by factoring the numerator.
1. Factor the quadratic expression \(x^2 + 5x + 6\). We look for two numbers that multiply to \(6\) (the constant term) and add up to \(5\) (the coefficient of \(x\)).
- The numbers \(2\) and \(3\) satisfy this condition because \(2 \cdot 3 = 6\) and \(2 + 3 = 5\).
So, we can factor \(x^2 + 5x + 6\) as:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
2. Now rewrite the original expression using this factorization:
\[
\frac{x^2 + 5x + 6}{x + 2} = \frac{(x + 2)(x + 3)}{x + 2}
\]
3. We see that \((x + 2)\) is common in the numerator and the denominator, so we can cancel it out (as long as \(x \neq -2\) to avoid division by zero):
\[
= x + 3 \quad \text{for } x \neq -2
\]
Therefore, the simplified expression is:
\[
x + 3 \quad \text{(for } x \neq -2\text{)}
\]
1. Factor the quadratic expression \(x^2 + 5x + 6\). We look for two numbers that multiply to \(6\) (the constant term) and add up to \(5\) (the coefficient of \(x\)).
- The numbers \(2\) and \(3\) satisfy this condition because \(2 \cdot 3 = 6\) and \(2 + 3 = 5\).
So, we can factor \(x^2 + 5x + 6\) as:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
2. Now rewrite the original expression using this factorization:
\[
\frac{x^2 + 5x + 6}{x + 2} = \frac{(x + 2)(x + 3)}{x + 2}
\]
3. We see that \((x + 2)\) is common in the numerator and the denominator, so we can cancel it out (as long as \(x \neq -2\) to avoid division by zero):
\[
= x + 3 \quad \text{for } x \neq -2
\]
Therefore, the simplified expression is:
\[
x + 3 \quad \text{(for } x \neq -2\text{)}
\]