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(x^3+27)/(x+3) Synthetic Division

3 min read Jun 17, 2024
(x^3+27)/(x+3) Synthetic Division

Synthetic Division: A Step-by-Step Guide for (x^3 + 27) / (x + 3)

Synthetic division is a handy shortcut for dividing polynomials, especially when the divisor is a linear expression of the form (x - a). Let's see how to apply this method to the problem (x^3 + 27) / (x + 3).

Step 1: Set up the problem

First, write down the coefficients of the dividend (x^3 + 27). Notice that the polynomial is missing the x^2 and x terms. We need to include these with a coefficient of 0.

The divisor is (x + 3), which means a = -3.

 -3 | 1  0  0  27
       ------------------

Step 2: Bring down the leading coefficient

Bring down the first coefficient (1) below the line.

 -3 | 1  0  0  27
       ------------------
           1

Step 3: Multiply and add

Multiply the number you just brought down (1) by the divisor (-3), and write the result (-3) below the next coefficient (0).

 -3 | 1  0  0  27
       ------------------
           1 -3

Add the numbers in the second column (0 and -3) and write the sum (-3) below the line.

 -3 | 1  0  0  27
       ------------------
           1 -3

Step 4: Repeat steps 3 and 4

Repeat the multiplication and addition steps for the remaining coefficients.

  • Multiply -3 by -3 to get 9.
  • Add 9 and 0 to get 9.
  • Multiply -3 by 9 to get -27.
  • Add -27 and 27 to get 0.
 -3 | 1  0  0  27
       ------------------
           1 -3  9  0

Step 5: Interpret the result

The numbers below the line represent the coefficients of the quotient, starting with the term of one degree less than the original polynomial. The last number is the remainder.

Therefore, (x^3 + 27) / (x + 3) = x^2 - 3x + 9 with a remainder of 0. This means that (x + 3) is a factor of (x^3 + 27).

Let's Summarize

Synthetic division provides a straightforward way to divide polynomials, especially when dealing with linear divisors. By following these steps, you can efficiently obtain the quotient and remainder for your polynomial division.

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