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 \(x - 2\), we will use \(2\) (the root of \(x - 2 = 0\)) in synthetic division.

Here are the steps of the synthetic division:

1. Write down the coefficients of the polynomial:
- For \(2x^3\), the coefficient is \(2\).
- For \(3x^2\), the coefficient is \(3\).
- For \(-21x\), the coefficient is \(-21\).
- For the constant term \(20\), the coefficient is \(20\).

So we have the coefficients: \(2, 3, -21, 20\).

2. Set up for synthetic division:

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

- Bring down the \(2\).
- Multiply \(2\) by \(2\) (the value being used in synthetic division) and write \(4\) underneath the next coefficient \(3\).
- Add \(3 + 4 = 7\).
- Multiply \(2\) by \(7\), and write \(14\) underneath \(-21\).
- Add \(-21 + 14 = -7\).
- Multiply \(2\) by \(-7\), and write \(-14\) underneath \(20\).
- Add \(20 - 14 = 6\).

3. The result of the synthetic division is:
- The coefficients of the quotient polynomial \(2x^2 + 7x - 7\).
- The remainder \(6\).

So, you can express the result of the division as:

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

Thus, the final result is:

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

Therefore, the complete expression is:

\[
2x^2 + 7x - 7 + \frac{6}{x-2}.
\]