Question

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:

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