How do you find the quotient of the polynomial 3x^3 + 4x + 32x^2 using synthetic division?

by

Synthetic division is a simplified method of dividing polynomials, particularly useful for dividing by linear factors. To find the quotient of the polynomial 3x3 + 32x2 + 4x using synthetic division, follow these steps:

Step 1: Set Up the Synthetic Division

First, we need to organize the coefficients of the polynomial:

  • For 3x3: the coefficient is 3
  • For 32x2: the coefficient is 32
  • For 4x: the coefficient is 4
  • For the constant term (which is 0): the coefficient is 0.

So, we write the coefficients: [3, 32, 4, 0].

Step 2: Choose the Value for Synthetic Division

In synthetic division, we typically divide by a linear factor like x – r. The value of r will depend on the root we are using. Let’s assume we are dividing by x – 4 (hence, r = 4). We will set up synthetic division using 4:

Step 3: Perform Synthetic Division

Now, we write 4 to the left and the coefficients to the right:

4 |  3    32    4    0
  |        ?    ?    ?

Now we bring down the 3:

4 |  3    32    4    0
  |        ?    ?    ?
  |--------------------
  |  3

Next, we multiply 3 by 4 and add it to the next coefficient:

4 |  3    32    4    0
  |        12
  |--------------------
  |  3    44

Continue the process:

4 |  3    32    4    0
  |        12    176
  |--------------------
  |  3    44   180

Finally, multiply 180 by 4 and perform the last addition:

4 |  3    32    4    0
  |        12    176       720
  |---------------------
  |  3    44    180   720

Step 4: Write the Result

The last row gives us the coefficients of the quotient and the remainder:

  • Quotient: 3x2 + 44x + 180
  • Remainder: 720

Thus, the quotient of the polynomial 3x3 + 32x2 + 4x divided by x – 4 is:

3x2 + 44x + 180 with a remainder of 720.

Conclusion

Using synthetic division simplifies the process of polynomial long division, providing a quick way to obtain the quotient and remainder efficiently. If you have any more questions about synthetic division or polynomials, feel free to ask!

Leave a Comment