Milind and Vineeth are two students from class IX. While preparing for their Mathematics exam, they approached Vineeth’s uncle to clear their doubts on the topic ‘Polynomials’. He used a cuboid to explain some important concepts. The volume of the cuboid is f(x) = x3 + 3x2 + ax + b and its length is (x – 1) and width is (x + 1).

(i)	f(1) = _____________
a) –1                   b) – 2                  c) 0                      d) 1

(ii)	   A cubic polynomial has _______________ zeroes.
(iii)	Find the values of a and b.
(iv)	 Find the height of the cuboid.

To solve the problem, let's go step by step.

### Given:
- The volume of the cuboid is \( f(x) = x^3 + 3x^2 + ax + b \).
- The length of the cuboid is \( x - 1 \).
- The width of the cuboid is \( x + 1 \).

Since the volume of the cuboid is given by the product of its length, width, and height, we can express the height \( h(x) \) as:
\[
h(x) = \frac{f(x)}{\text{length} \times \text{width}} = \frac{x^3 + 3x^2 + ax + b}{(x - 1)(x + 1)}
\]

Let's solve each part:

### Part (i) Find \( f(1) \):
Substitute \( x = 1 \) into the polynomial \( f(x) = x^3 + 3x^2 + ax + b \):
\[
f(1) = 1^3 + 3(1^2) + a(1) + b = 1 + 3 + a + b = 4 + a + b
\]
The value of \( f(1) \) will depend on the values of \( a \) and \( b \), which we haven't found yet. Let's hold off on this until we determine \( a \) and \( b \).

### Part (ii) Number of zeroes of a cubic polynomial:
A cubic polynomial always has **3 zeroes** (which can be real or complex).

### Part (iii) Find the values of \( a \) and \( b \):
To find \( a \) and \( b \), let's expand \( f(x) \) by multiplying the given factors of length, width, and height:
\[
(x - 1)(x + 1) = x^2 - 1
\]
Thus, the height \( h(x) \) is:
\[
h(x) = \frac{x^3 + 3x^2 + ax + b}{x^2 - 1}
\]
For \( f(x) \) to be divisible by \( x^2 - 1 \), the remainder must be 0 when \( f(x) \) is divided by \( x^2 - 1 \).

We can find \( a \) and \( b \) by equating the remainders to zero when dividing \( f(x) \) by \( x^2 - 1 \).

### Division:
Let's use polynomial long division:
1. Divide the leading term \( x^3 \) by \( x^2 \) to get \( x \).
2. Multiply \( x \) by \( x^2 - 1 \) to get \( x^3 - x \).
3. Subtract \( x^3 - x \) from \( x^3 + 3x^2 + ax + b \), giving \( 3x^2 + (a+1)x + b \).

Now, divide \( 3x^2 \) by \( x^2 \) to get \( 3 \).
1. Multiply \( 3 \) by \( x^2 - 1 \) to get \( 3x^2 - 3 \).
2. Subtract \( 3x^2 - 3 \) from \( 3x^2 + (a+1)x + b \) to get \( (a+1)x + (b+3) \).

For \( f(x) \) to be divisible by \( x^2 - 1 \), the remainder must be zero:
\[
(a+1)x + (b+3) = 0
\]
This implies:
\[
a + 1 = 0 \quad \text{and} \quad b + 3 = 0
\]
Thus, \( a = -1 \) and \( b = -3 \).

### Part (i) Revisited: Find \( f(1) \)
Now that we know \( a = -1 \) and \( b = -3 \), substitute them back into \( f(1) \):
\[
f(1) = 4 + a + b = 4 - 1 - 3 = 0
\]
So, \( f(1) = 0 \). The correct option is **(c) 0**.

### Part (iv) Find the height of the cuboid:
Since \( h(x) = \frac{f(x)}{(x-1)(x+1)} \), we substitute \( f(x) \) with the values of \( a \) and \( b \):
\[
f(x) = x^3 + 3x^2 - x - 3
\]
So,
\[
h(x) = \frac{x^3 + 3x^2 - x - 3}{x^2 - 1}
\]
Divide:
\[
h(x) = x + 3
\]
Thus, the height of the cuboid is \( h(x) = x + 3 \).

---

Would you like any further details or have any questions?

Here are five related questions:
1. How would the values of \( a \) and \( b \) change if the width was \( x - 1 \) instead?
2. What would be the effect on the height of the cuboid if the length were \( x + 2 \) instead?
3. How can you determine the zeroes of a cubic polynomial using factorization?
4. If \( f(x) = x^3 + 2x^2 + ax + b \) and the length is \( x - 2 \), how would you find \( a \) and \( b \)?
5. How does the degree of a polynomial relate to the maximum number of zeroes it can have?

**Tip**: Always remember that the degree of a polynomial indicates the maximum number of zeroes it can have, considering both real and complex roots.

Explore how polynomials and division relate to cuboid volume in mathematics. Learn to solve cubic equations and find polynomial coefficients to determine cuboid dimensions. Suitable for Grade 9 students.

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