On dividing the polynomial `x^(3)-5x^(2)+6x-4` by a polynomial `g(x)`, quotient and remainder are `(x-3)` and `(-3x+5)` respectively. Find `g(x)`

Answer

To solve the problem of finding the polynomial \( g(x) \) when dividing the polynomial \( p(x) = x^3 - 5x^2 + 6x - 4 \) by \( g(x) \), we can use the polynomial division algorithm. The algorithm states that:

\[ p(x) = g(x) \cdot q(x) + r(x) \]

where:
- \( p(x) \) is the dividend,
- \( g(x) \) is the divisor,
- \( q(x) \) is the quotient,
- \( r(x) \) is the remainder.

Given:
- \( p(x) = x^3 - 5x^2 + 6x - 4 \)
- \( q(x) = x - 3 \)
- \( r(x) = -3x + 5 \)

### Step 1: Set up the equation using the division algorithm

We can rewrite the equation as follows:

\[ x^3 - 5x^2 + 6x - 4 = g(x) \cdot (x - 3) + (-3x + 5) \]

### Step 2: Rearrange the equation

Rearranging gives:

\[ x^3 - 5x^2 + 6x - 4 + 3x - 5 = g(x) \cdot (x - 3) \]

This simplifies to:

\[ x^3 - 5x^2 + 9x - 9 = g(x) \cdot (x - 3) \]

### Step 3: Solve for \( g(x) \)

Now we can express \( g(x) \) as:

\[ g(x) = \frac{x^3 - 5x^2 + 9x - 9}{x - 3} \]

### Step 4: Perform polynomial long division

Now we will divide \( x^3 - 5x^2 + 9x - 9 \) by \( x - 3 \).

1. Divide the leading term: \( \frac{x^3}{x} = x^2 \).
2. Multiply \( x^2 \) by \( x - 3 \): \( x^2(x - 3) = x^3 - 3x^2 \).
3. Subtract:
\[ (x^3 - 5x^2) - (x^3 - 3x^2) = -5x^2 + 3x^2 = -2x^2 \]
4. Bring down the next term \( +9x \):
\[ -2x^2 + 9x \]
5. Divide the leading term: \( \frac{-2x^2}{x} = -2x \).
6. Multiply \( -2x \) by \( x - 3 \): \( -2x(x - 3) = -2x^2 + 6x \).
7. Subtract:
\[ (-2x^2 + 9x) - (-2x^2 + 6x) = 9x - 6x = 3x \]
8. Bring down the next term \( -9 \):
\[ 3x - 9 \]
9. Divide the leading term: \( \frac{3x}{x} = 3 \).
10. Multiply \( 3 \) by \( x - 3 \): \( 3(x - 3) = 3x - 9 \).
11. Subtract:
\[ (3x - 9) - (3x - 9) = 0 \]

### Final Result

Since the remainder is \( 0 \), we have:

\[ g(x) = x^2 - 2x + 3 \]

### Summary

Thus, the polynomial \( g(x) \) is:

\[ \boxed{x^2 - 2x + 3} \]