Question

Find the quotient and remainder using synthetic divison for : 2x63-14x^2+7x-32/ 2x^2 +5
The quotient is
The remainder is

Answer 1

To use synthetic division for the polynomial \(2x^3 - 14x^2 + 7x - 32\) divided by \(2x^2 + 5\), we need to adjust our approach since synthetic division is usually applicable for linear divisors. The polynomial we are dividing by is quadratic, \(2x^2 + 5\).

Instead of synthetic division, we can use polynomial long division.

### Step 1: Set up the division

We want to divide:

\[
\frac{2x^3 - 14x^2 + 7x - 32}{2x^2 + 5}
\]

### Step 2: Long Division Process

1. **Divide the leading term** of the dividend by the leading term of the divisor:

\[
\frac{2x^3}{2x^2} = x
\]

2. **Multiply** \(x\) by the entire divisor \(2x^2 + 5\):

\[
x(2x^2 + 5) = 2x^3 + 5x
\]

3. **Subtract** this result from the original polynomial:

\[
(2x^3 - 14x^2 + 7x - 32) - (2x^3 + 5x) = -14x^2 + 7x - 5x - 32 = -14x^2 + 2x - 32
\]

4. **Next, repeat the process:** Divide the leading term \(-14x^2\) by \(2x^2\):

\[
\frac{-14x^2}{2x^2} = -7
\]

5. **Multiply** \(-7\) by the entire divisor:

\[
-7(2x^2 + 5) = -14x^2 - 35
\]

6. **Subtract** this from the current polynomial:

\[
(-14x^2 + 2x - 32) - (-14x^2 - 35) = 2x - 32 + 35 = 2x + 3
\]

### Step 3: Collect results

We have divided the polynomial completely, and the results are:

- **Quotient:** \(x - 7\)
- **Remainder:** \(2x + 3\)

### Final Answer

Thus, the quotient is \(x - 7\) and the remainder is \(2x + 3\).