The area of a rectangles is `x^(3) - 8x^(2) + 7` and one of its sides ix x - 1. Find the length of the adjacent side.

Answer

To find the length of the adjacent side of the rectangle, we start with the information given:

1. The area of the rectangle is \( A = x^3 - 8x^2 + 7 \).
2. One of its sides (let's call it the breadth) is \( b = x - 1 \).

We need to find the length \( l \) of the adjacent side using the formula for the area of a rectangle:

\[ A = l \times b \]

### Step 1: Set up the equation
Using the area formula, we can express the length as:

\[ l = \frac{A}{b} = \frac{x^3 - 8x^2 + 7}{x - 1} \]

### Step 2: Perform polynomial long division
Now we will divide \( x^3 - 8x^2 + 7 \) by \( x - 1 \).

1. Divide the leading term of the numerator \( x^3 \) by the leading term of the denominator \( x \):
\[ x^3 \div x = x^2 \]
So, the first term of the quotient is \( x^2 \).

2. Multiply \( x^2 \) by \( x - 1 \):
\[ x^2(x - 1) = x^3 - x^2 \]

3. Subtract this from the original polynomial:
\[ (x^3 - 8x^2 + 7) - (x^3 - x^2) = -8x^2 + x^2 + 7 = -7x^2 + 7 \]

4. Now, divide the leading term of the new polynomial \( -7x^2 \) by \( x \):
\[ -7x^2 \div x = -7x \]
So, the next term of the quotient is \( -7x \).

5. Multiply \( -7x \) by \( x - 1 \):
\[ -7x(x - 1) = -7x^2 + 7x \]

6. Subtract this from the previous result:
\[ (-7x^2 + 7) - (-7x^2 + 7x) = 7 - 7x \]

7. Now, divide \( 7 - 7x \) by \( x - 1 \):
\[ -7x \div x = -7 \]
So, the last term of the quotient is \( -7 \).

8. Multiply \( -7 \) by \( x - 1 \):
\[ -7(x - 1) = -7x + 7 \]

9. Subtract this from the previous result:
\[ (7 - 7x) - (-7x + 7) = 0 \]

### Step 3: Write the final result
After performing the division, we find that:

\[ l = x^2 - 7x - 7 \]

Thus, the length of the adjacent side is:

\[ \boxed{x^2 - 7x - 7} \]