use long division to divide the polynomials 4x^3-21x^2+27x over x-3

Answer 1

To divide the polynomial \( 4x^3 - 21x^2 + 27x \) by \( x - 3 \) using long division, follow these steps:

1. Set up the division: Write \( 4x^3 - 21x^2 + 27x \) under the long division symbol and \( x - 3 \) on the outside.

           __________________________
   x - 3 | 4x^3 - 21x^2 + 27x
   

2. Divide the first term: Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( x \):

   \[ \frac{4x^3}{x} = 4x^2 \]

3. Multiply and subtract: Multiply the entire divisor \( x - 3 \) by \( 4x^2 \), and write the result underneath the dividend:

   \[ 4x^2(x - 3) = 4x^3 - 12x^2 \]

   Now perform the subtraction:

   \[ (4x^3 - 21x^2) - (4x^3 - 12x^2) = -21x^2 + 12x^2 = -9x^2 \]

   Bring down the next term \( +27x \):

           4x^2
           __________________________
   x - 3 | 4x^3 - 21x^2 + 27x
           | -(4x^3 - 12x^2)
           ----------------------
                    -9x^2 + 27x
   

4. Repeat the process: Now divide the first term of the new polynomial \( -9x^2 \) by \( x \):

   \[ \frac{-9x^2}{x} = -9x \]

   Multiply the divisor \( x - 3 \) by \( -9x \):

   \[ -9x(x - 3) = -9x^2 + 27x \]

   Perform the subtraction:

   \[ (-9x^2 + 27x) - (-9x^2 + 27x) = 0 \]

   So we have:

           4x^2 - 9x
           __________________________
   x - 3 | 4x^3 - 21x^2 + 27x
           | -(4x^3 - 12x^2)
           | ----------------------
                     -9x^2 + 27x
           | -(-9x^2 + 27x)
           | ----------------------
                          0
   

5. Result: Since there is no remainder, the division is complete. The quotient is:

\[ \boxed{4x^2 - 9x} \]