Question

Use long division to divide the polynomia -30^6 + 48x^5 +50x - 80 by 5x - 8 standard form

Answer 1

To divide the polynomial \(-30x^6 + 48x^5 + 50x - 80\) by the polynomial \(5x - 8\) using long division, we'll follow the standard polynomial long division steps.

### Step 1: Set up the long division

We write \(-30x^6 + 48x^5 + 0x^4 + 50x - 80\) under the long division bar and \(5x - 8\) outside, as follows:

```
___________________________
5x - 8 | -30x^6 + 48x^5 + 0x^4 + 50x - 80
```

### Step 2: Divide the first terms

Divide the leading term of the dividend \(-30x^6\) by the leading term of the divisor \(5x\):

\[
\frac{-30x^6}{5x} = -6x^5
\]

### Step 3: Multiply and subtract

Now we multiply \(-6x^5\) by \(5x - 8\) and subtract this result from the original polynomial:

\[
-6x^5 \cdot (5x - 8) = -30x^6 + 48x^5
\]

Now we subtract:

```
-6x^5
___________________________
5x - 8 | -30x^6 + 48x^5 + 0x^4 + 50x - 80
- (-30x^6 + 48x^5)
___________________
0x^5 + 0x^4 + 50x - 80
```

You end up with:

\[
0 + 0x^4 + 50x - 80
\]

### Step 4: Bring down

Next, we bring down the next term if necessary. In this case, we already have \(50x - 80\).

### Step 5: Repeat the process

Now we divide \(50x\) by \(5x\):

\[
\frac{50x}{5x} = 10
\]

Now we multiply \(10\) by \(5x - 8\):

\[
10 \cdot (5x - 8) = 50x - 80
\]

Now we subtract:

```
-6x^5 + 10
___________________________
5x - 8 | -30x^6 + 48x^5 + 0x^4 + 50x - 80
- (-30x^6 + 48x^5)
___________________
0 + 0x^4 + 50x - 80
- (50x - 80)
___________________
0
```

### Final Result

Since the remainder is \(0\), we found that:

\[
\frac{-30x^6 + 48x^5 + 50x - 80}{5x - 8} = -6x^5 + 10
\]

Thus, the final result of the division is:

\[
-6x^5 + 10
\]