Find the values of 'a' and ''b so that the polynomial `x^3+ax^2+bx-45` has (x-1) and (x+5) as its factors.
For the value sof 'a' and 'b', as obtained above, factorise the given polynomial completely.

Answer

To solve the problem, we need to find the values of 'a' and 'b' such that the polynomial \( P(x) = x^3 + ax^2 + bx - 45 \) has \( (x-1) \) and \( (x+5) \) as factors.

### Step 1: Use the Remainder Theorem
Since \( (x-1) \) is a factor, we can substitute \( x = 1 \) into the polynomial and set it equal to zero:
\[ P(1) = 1^3 + a(1^2) + b(1) - 45 = 0 \]
This simplifies to:
\[ 1 + a + b - 45 = 0 \]
Rearranging gives us:
\[ a + b - 44 = 0 \quad \text{(Equation 1)} \]

### Step 2: Use the Remainder Theorem for the second factor
Now, since \( (x+5) \) is also a factor, we substitute \( x = -5 \) into the polynomial and set it equal to zero:
\[ P(-5) = (-5)^3 + a(-5)^2 + b(-5) - 45 = 0 \]
This simplifies to:
\[ -125 + 25a - 5b - 45 = 0 \]
Rearranging gives us:
\[ 25a - 5b - 170 = 0 \]
Dividing the entire equation by 5 gives us:
\[ 5a - b - 34 = 0 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( a + b = 44 \)
2. \( 5a - b = 34 \)

We can solve these equations simultaneously. Adding both equations eliminates \( b \):
\[ (a + b) + (5a - b) = 44 + 34 \]
This simplifies to:
\[ 6a = 78 \implies a = 13 \]

Now substituting \( a = 13 \) back into Equation 1:
\[ 13 + b = 44 \implies b = 44 - 13 = 31 \]

### Step 4: Values of 'a' and 'b'
Thus, we have:
\[ a = 13, \quad b = 31 \]

### Step 5: Factor the polynomial completely
Now we substitute \( a \) and \( b \) back into the polynomial:
\[ P(x) = x^3 + 13x^2 + 31x - 45 \]

We can factor this polynomial using the known factors \( (x-1) \) and \( (x+5) \):
\[ P(x) = (x - 1)(x + 5)(\text{third factor}) \]

To find the third factor, we can perform polynomial long division or synthetic division, but since we know the factors, we can directly find the remaining factor.

### Step 6: Finding the third factor
We can express the polynomial as:
\[ P(x) = (x - 1)(x + 5)(x + c) \]
To find \( c \), we can expand:
\[ (x - 1)(x + 5) = x^2 + 4x - 5 \]
Now we multiply this by \( (x + c) \):
\[ (x^2 + 4x - 5)(x + c) = x^3 + (4+c)x^2 + (-5 + 4c)x - 5c \]
Setting this equal to \( x^3 + 13x^2 + 31x - 45 \), we can compare coefficients:
1. \( 4 + c = 13 \implies c = 9 \)
2. \( -5 + 4c = 31 \implies -5 + 36 = 31 \) (which is consistent)
3. \( -5c = -45 \implies c = 9 \)

Thus, the polynomial factors completely as:
\[ P(x) = (x - 1)(x + 5)(x + 9) \]

### Final Answer
The values of \( a \) and \( b \) are:
\[ a = 13, \quad b = 31 \]
The complete factorization of the polynomial is:
\[ P(x) = (x - 1)(x + 5)(x + 9) \]