Use the long division method to find the result when 2, x, cubed, plus, 7, x, squared, plus, 19, x, plus, 82x

Question

3
+7x
2
+19x+8 is divided by 2, x, plus, 12x+1.

Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 5, x, squared, plus, 5, x, plus, 1
4x
3
+5x
2
+5x+1 is divided by
4, x, plus, 1
4x+1.
Use Long Division:
Use Long Division:

x, squared
x
2
plus, x
+x
plus, 1
+1

4, x
4x
plus, 1
+1
right parenthesis
)
4, x, cubed
4x
3
plus, 5, x, squared
+5x
2
plus, 5, x
+5x
plus, 1
+1

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

4, x, squared
4x
2
plus, 5, x
+5x
plus, 1
+1

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

4, x
4x
plus, 1
+1

start superscript, minus, end superscript, 4, x
−
4x
plus, start superscript, minus, end superscript, 1
+
−
1

0
0

Answer: , x, squared, plus, x, plus, 1
Answer: x
2
+x+1

Answer 1

To divide \(2x^3 + 7x^2 + 19x + 82\) by \(2x + 12\) using long division, we will follow the method step-by-step.

Long Division

1. Setup the Long Division:

   • Divide \(2x^3 + 7x^2 + 19x + 82\) by \(2x + 12\).

                     ________________
   2x + 12 | 2x^3 + 7x^2 + 19x + 82
   

2. Divide the leading terms:

   • Divide \(2x^3\) by \(2x\) to get \(x^2\).

                     x^2
                     ________________
   2x + 12 | 2x^3 + 7x^2 + 19x + 82
   

3. Multiply and subtract:

   • Multiply \(x^2\) by \(2x + 12\) to get \(2x^3 + 12x^2\).
   • Subtract this from \(2x^3 + 7x^2\):

                     x^2
                     ________________
   2x + 12 | 2x^3 + 7x^2 + 19x + 82
                     -(2x^3 + 12x^2)
                     ________________
                             -5x^2 + 19x
   

4. Bring down the next term:

   • Bring down the \(+82\) to get \(-5x^2 + 19x + 82\).

                     x^2
                     ________________
   2x + 12 | 2x^3 + 7x^2 + 19x + 82
                     -(2x^3 + 12x^2)
                     ________________
                             -5x^2 + 19x + 82
   

5. Divide the leading terms again:

   • Divide \(-5x^2\) by \(2x\) to get \(-\frac{5}{2}x\).

                     x^2 - (5/2)x
                     ________________
   2x + 12 | 2x^3 + 7x^2 + 19x + 82
                     -(2x^3 + 12x^2)
                     ________________
                             -5x^2 + 19x + 82
                            -(-5x^2 - 30x)
                     ______________________
                                   49x + 82
   

6. Multiply and subtract:

   • Multiply \(-\frac{5}{2}x\) by \(2x + 12\) to get \(-5x^2 - 30x\).
   • Subtract this from the previous remainder:

                                  49x + 82
                            - (-5x^2 - 30x)
                     ______________________
                                  49x + 82
                            - (49x + 588)
                     _______________________ 
                                             -506
   

7. Divide the constant:

   • Divide \(49x\) by \(2x\) to get \(24.5\), but for simplicity, we will keep it in the form we've got.

8. Final Result:

   • The final step gives us the remainder which is \(-506\).
   • Hence, we can write the answer:

Answer: x^2 - (5/2)x + 24.5 with a remainder of -506

Combining everything, you can represent the answer as:

Answer: \(x^2 - \frac{5}{2}x + 24.5 - \frac{506}{2x + 12}\)