If the polynomial `4x^(3)-16x^(2)+ax+7` is exactly divisible by x-1, then find the value of a. Hence factorise the polynomial.

Answer

To solve the problem, we need to find the value of \( a \) such that the polynomial \( 4x^3 - 16x^2 + ax + 7 \) is exactly divisible by \( x - 1 \). Then, we will factorize the polynomial.

### Step 1: Use the Remainder Theorem
Since \( x - 1 \) is a factor of the polynomial, by the Remainder Theorem, the value of the polynomial at \( x = 1 \) should be equal to 0.

### Step 2: Substitute \( x = 1 \) into the polynomial
Let's substitute \( x = 1 \) into the polynomial:

\[ p(1) = 4(1)^3 - 16(1)^2 + a(1) + 7 \]

Calculating each term:

\[ p(1) = 4(1) - 16(1) + a + 7 \]
\[ = 4 - 16 + a + 7 \]
\[ = (4 - 16 + 7) + a \]
\[ = -5 + a \]

### Step 3: Set the polynomial equal to 0
Since \( p(1) = 0 \):

\[ -5 + a = 0 \]

### Step 4: Solve for \( a \)
Now, we can solve for \( a \):

\[ a = 5 \]

### Step 5: Substitute \( a \) back into the polynomial
Now that we have found \( a \), we substitute it back into the polynomial:

\[ p(x) = 4x^3 - 16x^2 + 5x + 7 \]

### Step 6: Factor the polynomial
Now we will factor \( p(x) \) using synthetic division or polynomial long division by \( x - 1 \).

#### Synthetic Division Steps:
1. Write down the coefficients: \( 4, -16, 5, 7 \).
2. Use \( 1 \) (the root from \( x - 1 = 0 \)) for synthetic division.

```
1 | 4 -16 5 7
| 4 -12 -7
-------------------
4 -12 -7 0
```

The result of the synthetic division is \( 4x^2 - 12x - 7 \).

### Step 7: Factor \( 4x^2 - 12x - 7 \)
Now we need to factor \( 4x^2 - 12x - 7 \). We can use the middle-term splitting method.

1. The product of the coefficient of \( x^2 \) (which is \( 4 \)) and the constant term (which is \( -7 \)) is \( -28 \).
2. We need two numbers that multiply to \( -28 \) and add to \( -12 \). These numbers are \( -14 \) and \( 2 \).

So we can rewrite \( -12x \) as \( -14x + 2x \):

\[ 4x^2 - 14x + 2x - 7 \]

### Step 8: Group and factor
Now, we group the terms:

\[ (4x^2 - 14x) + (2x - 7) \]

Factoring out the common factors:

\[ 2x(2x - 7) + 1(2x - 7) \]

Now factor out \( (2x - 7) \):

\[ (2x - 7)(2x + 1) \]

### Final Factorization
Thus, the complete factorization of the polynomial \( 4x^3 - 16x^2 + 5x + 7 \) is:

\[ p(x) = (x - 1)(2x - 7)(2x + 1) \]

### Summary
- The value of \( a \) is \( 5 \).
- The factorization of the polynomial is \( (x - 1)(2x - 7)(2x + 1) \).