Question

find the quotient and remainder using synthetic division: \\(\frac{x^3 + 3x^2 - 19}{x - 2}\\)

the quotient is \\(x^2 + x - 2\\)

the remainder is \\(-15\\)

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shortcut method to divide polynomials by linear factors.

Step 1: Set up the synthetic division

To divide \( \frac{x^3 + 3x^2 - 19}{x - 2} \), we first identify the divisor's root and write down the coefficients of the dividend.

The divisor is \( x - 2 \), so the root we use is \( c = 2 \).

The dividend is \( x^3 + 3x^2 - 19 \). We must include a placeholder coefficient of \( 0 \) for the missing \( x \) term:

$$ x^3 + 3x^2 + 0x - 19 $$

The coefficients are:

$$ 1, \quad 3, \quad 0, \quad -19 $$

We set up the synthetic division table:

  2 |  1    3    0   -19
    |
    -------------------

Step 2: Perform the synthetic division steps

1. Bring down the first coefficient, \( 1 \):

  2 |  1    3    0   -19
    |
    -------------------
       1

1. Multiply \( 2 \) by \( 1 \) to get \( 2 \), write it under the \( 3 \), and add:

$$ 3 + 2 = 5 $$

  2 |  1    3    0   -19
    |       2
    -------------------
       1    5

1. Multiply \( 2 \) by \( 5 \) to get \( 10 \), write it under the \( 0 \), and add:

$$ 0 + 10 = 10 $$

  2 |  1    3    0   -19
    |       2   10
    -------------------
       1    5   10

1. Multiply \( 2 \) by \( 10 \) to get \( 20 \), write it under the \( -19 \), and add:

$$ -19 + 20 = 1 $$

  2 |  1    3    0   -19
    |       2   10    20
    -------------------
       1    5   10  |  1

Step 3: Interpret the results

The numbers at the bottom represent the coefficients of the quotient and the remainder.

• The last number, \( 1 \), is the remainder.
• The other numbers, \( 1, 5, 10 \), represent the coefficients of the quotient polynomial, which starts one degree lower than the original dividend (\( x^2 \)):

$$ 1x^2 + 5x + 10 $$

Answer:

• The quotient is \( x^2 + 5x + 10 \)
• The remainder is \( 1 \)