Question

How do you divide polynomials using the long division method?

Answer 1

Dividing polynomials using the long division method is similar to dividing numbers. Here’s a step-by-step guide on how to do it:

### Step 1: Set Up the Division

Write the dividend (the polynomial you want to divide) under the long division symbol and the divisor (the polynomial you are dividing by) outside.

### Example:

Let's divide \( 2x^3 + 3x^2 - 4x + 5 \) by \( x - 2 \).

```
____________
x - 2 | 2x^3 + 3x^2 - 4x + 5
```

### Step 2: Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor.

In our example:
- Divide \( 2x^3 \) by \( x \) to get \( 2x^2 \).

### Step 3: Multiply and Subtract

Multiply the entire divisor \( x - 2 \) by the result from Step 2 (which is \( 2x^2 \)) and write it beneath the corresponding terms of the dividend.

```
2x^2
____________
x - 2 | 2x^3 + 3x^2 - 4x + 5
-(2x^3 - 4x^2)
______________
7x^2 - 4x + 5
```

Now, subtract the result from the dividend. Make sure to change the signs:

- \( 3x^2 - (-4x^2) = 3x^2 + 4x^2 = 7x^2 \)

### Step 4: Bring Down the Next Term

Bring down the next term from the dividend to create a new polynomial.

Now we have \( 7x^2 - 4x \).

### Step 5: Repeat the Process

Repeat the process: divide the leading term of the current polynomial by the leading term of the divisor, multiply, and subtract.

Divide \( 7x^2 \) by \( x \) to get \( 7x \):

- Multiply \( 7x \) by \( x - 2 \):

```
2x^2 + 7x
____________
x - 2 | 2x^3 + 3x^2 - 4x + 5
-(2x^3 - 4x^2)
______________
7x^2 - 4x + 5
-(7x^2 - 14x)
_______________
10x + 5
```

### Step 6: Continue until Finished

Repeat again with \( 10x + 5 \):

- Divide \( 10x \) by \( x \) to get \( 10 \).
- Multiply \( 10 \) by \( x - 2 \):

```
2x^2 + 7x + 10
____________
x - 2 | 2x^3 + 3x^2 - 4x + 5
-(2x^3 - 4x^2)
______________
7x^2 - 4x + 5
-(7x^2 - 14x)
_______________
10x + 5
-(10x - 20)
_______________
25
```

### Step 7: Final Remainder

After completing all possible divisions, the final result will include the quotient and any remainder.

In this example, our quotient is \( 2x^2 + 7x + 10 \) with a remainder of \( 25 \).

### Final Answer:

So we conclude with:

\[
\frac{2x^3 + 3x^2 - 4x + 5}{x - 2} = 2x^2 + 7x + 10 + \frac{25}{x - 2}
\]

That's how you divide polynomials using the long division method!