Answer:
Step-by-step explanation:
To find the fourth proportional to \( x^2 - 5x + 4 \), \( x^2 + x - 2 \), and \( x^2 - 16 \), we need to find a number \( y \) such that these numbers form a proportion:
\[ \frac{x^2 - 5x + 4}{x^2 + x - 2} = \frac{x^2 - 16}{y} \]
Let's solve step by step:
1. **Set up the proportion:**
\[ \frac{x^2 - 5x + 4}{x^2 + x - 2} = \frac{x^2 - 16}{y} \]
2. **Cross-multiply to eliminate the fraction:**
\[ (x^2 - 5x + 4) \cdot y = (x^2 + x - 2) \cdot (x^2 - 16) \]
3. **Expand both sides:**
\[ y(x^2 - 5x + 4) = (x^2 + x - 2)(x^2 - 16) \]
4. **Multiply out the right-hand side:**
\[ (x^2 + x - 2)(x^2 - 16) = x^4 - 16x^2 + x^3 - 16x + x^2 - 2x - 2x^2 + 32 \]
\[ = x^4 + x^3 - 18x^2 - 18x + 32 \]
5. **Divide both sides by \( x^2 - 5x + 4 \) to solve for \( y \):**
\[ y = \frac{x^4 + x^3 - 18x^2 - 18x + 32}{x^2 - 5x + 4} \]
Therefore, the fourth proportional to \( x^2 - 5x + 4 \), \( x^2 + x - 2 \), and \( x^2 - 16 \) is \( \frac{x^4 + x^3 - 18x^2 - 18x + 32}{x^2 - 5x + 4} \).
