Dividing a Polynomial by Monomials & Binomials | Steps & Examples ... - Free Printable

Let’s solve this polynomial division step by step.

We are dividing:
Dividend: \(2x^2 - 6x + 3\)
Divisor: \(x - 2\)

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Step 1: Divide the first term of the dividend by the first term of the divisor.

First term of dividend: \(2x^2\)
First term of divisor: \(x\)
→ \(2x^2 ÷ x = 2x\) → This is the first term of the quotient.

Write “2x” above the division bar.

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Step 2: Multiply the entire divisor by this term.

Multiply \(2x \cdot (x - 2) = 2x^2 - 4x\)

Write this under the dividend and subtract:

```
2x
_______
x-2 | 2x² - 6x + 3
-(2x² - 4x)
___________
-2x + 3
```

Subtracting:
\(2x^2 - 6x + 3 - (2x^2 - 4x) = 0x^2 -2x + 3\)

So now we have remainder so far: \(-2x + 3\)

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Step 3: Repeat — divide the new leading term by the divisor’s leading term.

New leading term: \(-2x\)
Divide by \(x\): \(-2x ÷ x = -2\) → This is the second term of the quotient.

Add “-2” to the quotient → now quotient is \(2x - 2\)

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Step 4: Multiply divisor by this new term.

Multiply \(-2 \cdot (x - 2) = -2x + 4\)

Write this under the current remainder and subtract:

```
2x - 2
________
x - 2 | 2x² - 6x + 3
-(2x² - 4x)
___________
-2x + 3
-(-2x + 4) ← note: subtracting means change signs!
___________
-1
```

Wait — let’s be careful with subtraction:

We have:
Current line: \(-2x + 3\)
Minus: \((-2x + 4)\) → which is same as adding \(+2x - 4\)

So:
\(-2x + 3 - (-2x + 4) = -2x + 3 + 2x - 4 = 0x -1 = -1\)

✔ So the final remainder is -1

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Final Result:

Quotient: \(2x - 2\)
Remainder: \(-1\)

So, the answer can be written as:

\[
2x - 2 + \frac{-1}{x - 2} \quad \text{or} \quad 2x - 2 - \frac{1}{x - 2}
\]

But since the problem shows the long division setup and ends with remainder -1, and asks for the result of the division (as shown in the image), the expected answer is likely just stating the quotient and remainder.

However, looking at the image — it already completed the division and got remainder -1. The question might be asking to verify or state the result.

Since no explicit question was given beyond the image, but based on standard interpretation of such problems, we report:

> When \(2x^2 - 6x + 3\) is divided by \(x - 2\), the quotient is \(2x - 2\) and the remainder is \(-1\).

But if the task is simply to complete or confirm the division shown — then yes, it's correct.

Wait — actually, rechecking the arithmetic in the image:

In the second subtraction step:

It says:
After getting \(-2x + 3\), they write:
\(-(-2x + 4)\) → which becomes \(+2x - 4\), so:

\(-2x + 3 + 2x - 4 = -1\) → Correct.

And quotient is \(2x - 2\) → Correct.

So everything checks out.

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Final Answer:
The quotient is \(2x - 2\) and the remainder is \(-1\).