To determine the domain restrictions for the rational expression

$\frac{x^2 + x + 6}{x^2 + 2x - 3}$

we need to identify values of $x$ that would make the denominator zero, as division by zero is undefined.

1. Start with the denominator: $x^2 + 2x - 3 = 0$

2. Factor the quadratic expression: $x^2 + 2x - 3 = (x + 3)(x - 1) = 0$

3. Set each factor equal to zero to solve for $x$: $x + 3 = 0 \Rightarrow x = -3$ $x - 1 = 0 \Rightarrow x = 1$

Therefore, the values that make the denominator zero are $x = -3$ and $x = 1$. These are the values that must be excluded from the domain.

Answer

The correct domain restrictions for this rational expression are:

• $x \neq -3$
• $x \neq 1$

Correct Options

You should select:

• $x \neq -3$
• $x \neq 1$

Would you like a further breakdown, or do you have any questions?

Related Questions

1. What are domain restrictions in rational expressions, and why are they important?
2. How do you factor quadratic expressions like $x^2 + 2x - 3$?
3. Can the numerator affect the domain restrictions of a rational expression?
4. What if there were additional terms in the denominator? How would that affect the solution?
5. How can you verify the factors of a quadratic polynomial?

Tip

When factoring, check each factor by expanding it back to ensure accuracy in simplifying expressions.