Use the long division method to find the result when 4, x, cubed, minus, x, squared, minus, 13, x, plus, 104x

Question

3
−x
2
−13x+10 is divided by x, minus, 1x−1.

Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 12, x, squared, plus, 27, x, plus, 27
4x
3
+12x
2
+27x+27 is divided by
2, x, plus, 3
2x+3.
Use Long Division:
Use Long Division:

2, x, squared
2x
2
plus, 3, x
+3x
plus, 9
+9

2, x
2x
plus, 3
+3
right parenthesis
)
4, x, cubed
4x
3
plus, 12, x, squared
+12x
2
plus, 27, x
+27x
plus, 27
+27

start superscript, minus, end superscript, 4, x, cubed
−
4x
3
plus, start superscript, minus, end superscript, 6, x, squared
+
−
6x
2

6, x, squared
6x
2
plus, 27, x
+27x
plus, 27
+27

start superscript, minus, end superscript, 6, x, squared
−
6x
2
plus, start superscript, minus, end superscript, 9, x
+
−
9x

18, x
18x
plus, 27
+27

start superscript, minus, end superscript, 18, x
−
18x
plus, start superscript, minus, end superscript, 27
+
−

Answer 1

Let's perform long division of the polynomial \(4x^3 - x^2 - 13x + 104\) by \(x - 1\).

Long Division Steps:

1. Set up the long division:

             _______________________
   x - 1 | 4x^3 - x^2 - 13x + 104
   

2. Divide the first term:

   Divide \(4x^3\) by \(x\):

   • Result: \(4x^2\)

   Place \(4x^2\) above the long division line:

             4x^2
             _______________________
   x - 1 | 4x^3 - x^2 - 13x + 104
   

3. Multiply and subtract:

   Multiply \(4x^2\) by \(x - 1\):

   • Result: \(4x^3 - 4x^2\)

   Subtract this from the original polynomial:

             4x^2
             _______________________
   x - 1 | 4x^3 - x^2 - 13x + 104
             - (4x^3 - 4x^2)
             _______________________
                      3x^2 - 13x + 104
   

4. Next term:

   Divide \(3x^2\) by \(x\):

   • Result: \(3x\)

   Add this to the division result:

             4x^2 + 3x
             _______________________
   x - 1 | 4x^3 - x^2 - 13x + 104
             - (4x^3 - 4x^2)
             _______________________
                      3x^2 - 13x + 104
   

5. Multiply and subtract:

   Multiply \(3x\) by \(x - 1\):

   • Result: \(3x^2 - 3x\)

   Subtract:

             4x^2 + 3x
             _______________________
   x - 1 | 4x^3 - x^2 - 13x + 104
             - (4x^3 - 4x^2)
             _______________________
                      3x^2 - 13x + 104
                     - (3x^2 - 3x)
             _______________________
                             -10x + 104
   

6. Next term:

   Divide \(-10x\) by \(x\):

   • Result: \(-10\)

   Add this to the division result:

             4x^2 + 3x - 10
             _______________________
   x - 1 | 4x^3 - x^2 - 13x + 104
             - (4x^3 - 4x^2)
             _______________________
                      3x^2 - 13x + 104
                     - (3x^2 - 3x)
             _______________________
                             -10x + 104
                          - (-10x + 10)
             _______________________
                                     94
   

7. Final result:

Since we can't divide further (the degree of \(94\) is less than that of \(x - 1\)), we can write the complete answer. The quotient is:

\[ \boxed{4x^2 + 3x - 10} \quad \text{with a remainder of } \quad 94. \]

Thus, the answer is: \[ 4x^2 + 3x - 10 + \frac{94}{x - 1}. \]