If `h(y)=y^(2)andg(z)=z^(3)`, then the HCF of `h(b)-h(a) and g(b)-g(a)` is ___________.

Options

A. `a+b`
B. `a^(2)-b^(2)`
C. `b-a`
D. `a-b`

Answer

To find the HCF of \( h(b) - h(a) \) and \( g(b) - g(a) \), we start with the given functions:

1. **Define the Functions:**
- \( h(y) = y^2 \)
- \( g(z) = z^3 \)

2. **Calculate \( h(b) - h(a) \):**
- \( h(b) = b^2 \)
- \( h(a) = a^2 \)
- Therefore,
\[ h(b) - h(a) = b^2 - a^2 \]
- We can factor this using the difference of squares formula:
\[ b^2 - a^2 = (b - a)(b + a) \]

3. **Calculate \( g(b) - g(a) \):**
- \( g(b) = b^3 \)
- \( g(a) = a^3 \)
- Therefore,
\[ g(b) - g(a) = b^3 - a^3 \]
- We can factor this using the difference of cubes formula:
\[ b^3 - a^3 = (b - a)(b^2 + ab + a^2) \]

4. **Finding the HCF:**
- Now we have:
- \( h(b) - h(a) = (b - a)(b + a) \)
- \( g(b) - g(a) = (b - a)(b^2 + ab + a^2) \)
- The common factor in both expressions is \( (b - a) \).

5. **Conclusion:**
- Therefore, the HCF of \( h(b) - h(a) \) and \( g(b) - g(a) \) is:
\[ \text{HCF} = b - a \]