Certainly, let's solve the given problems step by step:
### Part (a): Find the length and breadth of the field.
The area of the rectangular field is given by the expression [tex]\( x^2 - 8x + 15 \)[/tex].
To find the dimensions (length and breadth) of the rectangular field, we need to factorize this quadratic expression:
[tex]\[ x^2 - 8x + 15 \][/tex]
We can factorize it by finding two numbers that multiply to 15 and add up to -8. The numbers -3 and -5 fit this requirement:
[tex]\[ x^2 - 8x + 15 = (x - 3)(x - 5) \][/tex]
Thus, the length and breadth of the field are:
- Length: [tex]\( x - 5 \)[/tex]
- Breadth: [tex]\( x - 3 \)[/tex]
### Part (b): Find the perimeter of the field.
Using the dimensions obtained above, we can find the perimeter of the rectangular field.
The formula for the perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[ P = 2 \times (\text{Length} + \text{Breadth}) \][/tex]
Substituting the dimensions:
[tex]\[ P = 2 \times \left((x - 5) + (x - 3)\right) \][/tex]
[tex]\[ P = 2 \times \left(x - 5 + x - 3\right) \][/tex]
[tex]\[ P = 2 \times (2x - 8) \][/tex]
[tex]\[ P = 4x - 16 \][/tex]
Therefore, the perimeter of the field is:
[tex]\[ 4x - 16 \][/tex]
### Part (c): Find the H.C.F. of [tex]\( a^2 - b^2 \)[/tex] and [tex]\( a^3 - b^3 \)[/tex].
To find the highest common factor (H.C.F.) of the expressions [tex]\( a^2 - b^2 \)[/tex] and [tex]\( a^3 - b^3 \)[/tex], we first recall their factorized forms.
1. The expression for [tex]\( a^2 - b^2 \)[/tex] (difference of squares) is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
2. The expression for [tex]\( a^3 - b^3 \)[/tex] (difference of cubes) is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Next, we identify the common factors between the two expressions:
- Both expressions share [tex]\( a - b \)[/tex] as a common factor.
Therefore, the H.C.F. of [tex]\( a^2 - b^2 \)[/tex] and [tex]\( a^3 - b^3 \)[/tex] is:
[tex]\[ a - b \][/tex]
### Summary of results:
a) The length of the field is [tex]\( x - 5 \)[/tex] and the breadth of the field is [tex]\( x - 3 \)[/tex].
b) The perimeter of the field is [tex]\( 4x - 16 \)[/tex].
c) The H.C.F. of [tex]\( a^2 - b^2 \)[/tex] and [tex]\( a^3 - b^3 \)[/tex] is [tex]\( a - b \)[/tex].
