(3x^3+17x^2+21x-9)/(x+3) Synthetic Division

Jasonbradley

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·

Jun 16, 2024

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Performing Synthetic Division: (3x^3 + 17x^2 + 21x - 9) / (x + 3)

Synthetic division is a simplified method for dividing polynomials, specifically when the divisor is a linear expression in the form of (x - a). In this case, we'll be dividing (3x^3 + 17x^2 + 21x - 9) by (x + 3).

Here's how to perform the synthetic division:

1. Set up the problem:

• Write the coefficients of the dividend (3x^3 + 17x^2 + 21x - 9) in a row.
• Write the value of 'a' from the divisor (x + 3) to the left. Since it's (x + 3), 'a' is -3.

-3 | 3  17  21  -9
     ----------------

2. Bring down the first coefficient:

• Bring down the first coefficient (3) below the line.

-3 | 3  17  21  -9
     ----------------
      3

3. Multiply and add:

• Multiply the number you just brought down (3) by the divisor's constant (-3). Write the result ( -9) under the next coefficient (17).
• Add the two numbers (17 + -9 = 8).

-3 | 3  17  21  -9
     ----------------
      3   -9
     --------
         8

4. Repeat steps 3 and 4:

• Multiply the new number (8) by the divisor's constant (-3) and write the result (-24) under the next coefficient (21).
• Add (21 + -24 = -3).

-3 | 3  17  21  -9
     ----------------
      3   -9  -24
     --------
         8   -3

• Repeat the process for the last coefficient: Multiply (-3) by -3 and add to -9.

-3 | 3  17  21  -9
     ----------------
      3   -9  -24   9
     --------
         8   -3   0

5. Interpret the results:

The numbers below the line (3, 8, -3) represent the coefficients of the quotient, with the last number (0) being the remainder.

Therefore, the result of the division (3x^3 + 17x^2 + 21x - 9) / (x + 3) is:

(3x^2 + 8x - 3) with a remainder of 0.

This means:

(3x^3 + 17x^2 + 21x - 9) = (x + 3)(3x^2 + 8x - 3)