Question

Use long division to divide 8x3−20x2+18x−11%0D%0A8%0D%0A𝑥%0D%0A3%0D%0A−%0D%0A20%0D%0A𝑥%0D%0A2%0D%0A+%0D%0A18%0D%0A𝑥%0D%0A−%0D%0A11%0D%0A by 2x2−3

Answer 1

To divide the polynomial \( 8x^3 - 20x^2 + 18x - 11 \) by \( 2x^2 - 3 \) using long division, we can follow these steps:

### Step 1: Set up the long division

We write \( 8x^3 - 20x^2 + 18x - 11 \) under the long division bracket and \( 2x^2 - 3 \) on the outside.

```
_________________________
2x^2 - 3 | 8x^3 - 20x^2 + 18x - 11
```

### Step 2: Divide the leading term

Now, we divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( 2x^2 \):

\[
\frac{8x^3}{2x^2} = 4x
\]

### Step 3: Multiply and subtract

Next, we multiply the entire divisor \( 2x^2 - 3 \) by \( 4x \):

\[
4x(2x^2 - 3) = 8x^3 - 12x
\]

Now we subtract this from the original polynomial:

\[
(8x^3 - 20x^2 + 18x - 11) - (8x^3 - 12x) = -20x^2 + 30x - 11
\]

### Step 4: Repeat

Now we repeat the process with the new polynomial \( -20x^2 + 30x - 11 \).

1. Divide the leading term \( -20x^2 \) by \( 2x^2 \):

\[
\frac{-20x^2}{2x^2} = -10
\]

2. Multiply the divisor \( 2x^2 - 3 \) by \( -10 \):

\[
-10(2x^2 - 3) = -20x^2 + 30
\]

3. Subtract from \( -20x^2 + 30x - 11 \):

\[
(-20x^2 + 30x - 11) - (-20x^2 + 30) = 30x - 11 - 30 = 30x - 41
\]

### Step 5: End the division

Now we take the result \( 30x - 41 \) and notice that the degree of \( 30x - 41 \) (which is 1) is less than the degree of the divisor \( 2x^2 - 3 \) (which is 2). Therefore, we cannot divide further.

### Conclusion

Putting it all together, the quotient and remainder from the division are:

\[
\text{Quotient} = 4x - 10
\]
\[
\text{Remainder} = 30x - 41
\]

Thus, we can express the result of the division as:

\[
\frac{8x^3 - 20x^2 + 18x - 11}{2x^2 - 3} = 4x - 10 + \frac{30x - 41}{2x^2 - 3}
\]