use synthetic division to find the result when 2x^3+3x^2-21x+20 is divided by x-2 if there is a remainder express the result is the form of q(x)+r(x)/b(x)

Answer 1

To perform synthetic division of the polynomial \( 2x^3 + 3x^2 - 21x + 20 \) by the linear polynomial \( x - 2 \), we will follow these steps:

1. Set up the synthetic division: We use the root of the divisor \( x - 2 \), which is \( 2 \). We list the coefficients of the polynomial: \( 2, 3, -21, 20 \).

2. Perform the synthetic division:

   • Write \( 2 \) to the left and the coefficients on the right:

     \[ \begin{array}{r|rrrr} 2 & 2 & 3 & -21 & 20 \ & & 4 & 14 & -14 \ \hline & 2 & 7 & -7 & 6 \ \end{array} \]

   • Steps:

     1. Bring down the leading coefficient \( 2 \).
     2. Multiply \( 2 \) by the number just written below the line (which is \( 2 \)), and write the result \( 4 \) under the next coefficient \( 3 \).
     3. Add \( 3 + 4 = 7 \).
     4. Multiply \( 2 \) by \( 7 \) to get \( 14 \) and write it under \( -21 \).
     5. Add \( -21 + 14 = -7 \).
     6. Multiply \( 2 \) by \( -7 \) to get \( -14 \) and write it under \( 20 \).
     7. Add \( 20 + (-14) = 6 \).

Thus, the results of the synthetic division produce the coefficients \( 2, 7, -7 \) for the quotient and \( 6 \) as the remainder.

3. Write the result: This means that

\[ 2x^3 + 3x^2 - 21x + 20 = (x - 2)(2x^2 + 7x - 7) + 6. \]

In the form \( q(x) + \frac{r(x)}{b(x)} \) (where \( a(x) \) is our quotient and \( r \) is the remainder):

\[ q(x) = 2x^2 + 7x - 7, \quad r(x) = 6, \quad b(x) = x - 2. \]

In conclusion, we can express the result as:

\[ 2x^3 + 3x^2 - 21x + 20 = (2x^2 + 7x - 7) + \frac{6}{x - 2}. \]