The expression you provided, $\frac{x^2 - 7x + 18}{x - 4}$, can be simplified by dividing the polynomial in the numerator by the polynomial in the denominator if possible.

Step-by-Step Solution:

1. Check for Factorization: We should first see if $x^2 - 7x + 18$ can be factored to allow us to cancel terms with the denominator $x - 4$.

2. Factor the Quadratic Expression: For the quadratic $x^2 - 7x + 18$, we look for two numbers that multiply to $18$ and add to $-7$. However, no real numbers satisfy both these conditions because $18$ does not have factors that add up to $-7$. Thus, this quadratic cannot be factored over the real numbers.

3. Long Division: Since factoring is not possible, we can proceed with polynomial long division to divide $x^2 - 7x + 18$ by $x - 4$.

   • Divide $x^2$ by $x$, which gives $x$.
   • Multiply $x$ by $x - 4$, yielding $x^2 - 4x$.
   • Subtract $(x^2 - 4x)$ from $x^2 - 7x + 18$, which results in $-3x + 18$.
   • Divide $-3x$ by $x$, yielding $-3$.
   • Multiply $-3$ by $x - 4$, resulting in $-3x + 12$.
   • Subtract $-3x + 12$ from $-3x + 18$, leaving a remainder of $6$.

   So, the result of the division is: $\frac{x^2 - 7x + 18}{x - 4} = x - 3 + \frac{6}{x - 4}$

Final Answer:

$\frac{x^2 - 7x + 18}{x - 4} = x - 3 + \frac{6}{x - 4}$

Would you like further details or have any questions?

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Related Questions

1. How can polynomial long division be applied to other polynomial functions?
2. What is the significance of remainders in polynomial division?
3. How does synthetic division differ from polynomial long division?
4. Can complex factors be used if factoring fails over real numbers?
5. How does dividing by a polynomial affect the degree of the result?

Tip:

When simplifying rational expressions, always check for common factors to cancel out before using long division.