Polynomial long division example demonstrating division of a cubic polynomial by a binomial.

Let’s solve the polynomial division:
(4x³ – 17x² + 13x – 41) ÷ (x – 4)

We’ll use polynomial long division, step by step.

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Step 1: Set up the division

Dividend: 4x³ – 17x² + 13x – 41
Divisor: x – 4

We write it like this:

```
________________
x - 4 ) 4x³ - 17x² + 13x - 41
```

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Step 2: Divide first term of dividend by first term of divisor

First term: 4x³ ÷ x = 4x²

Write 4x² on top.

Multiply 4x² × (x – 4) = 4x³ – 16x²

Subtract that from the dividend:

(4x³ – 17x²) – (4x³ – 16x²) = –x²

Bring down next term: +13x → now we have –x² + 13x

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Step 3: Repeat — divide –x² by x

–x² ÷ x = –x

Write –x on top (next to 4x²).

Multiply –x × (x – 4) = –x² + 4x

Subtract:

(–x² + 13x) – (–x² + 4x) = 9x

Bring down next term: –41 → now we have 9x – 41

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Step 4: Repeat — divide 9x by x

9x x = 9

Write +9 on top.

Multiply 9 × (x – 4) = 9x – 36

Subtract:

(9x – 41) – (9x – 36) = –5

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Final Result:

Quotient: 4x² – x + 9
Remainder: –5

So, the answer is:

> 4x² – x + 9 with a remainder of –5

Or written as:

> 4x² – x + 9 – 5/(x – 4)

But since the problem just says “÷”, and doesn’t specify format, we can leave it as quotient and remainder.

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✔ Let’s double-check our work by multiplying back:

(x – 4)(4x² – x + 9) + (-5) should equal original dividend.

First, multiply:

(x)(4x² – x + 9) = 4x³ – x² + 9x
(-4)(4x² – x + 9) = –16x² + 4x – 36

Add them:

4x³ – x² + 9x –16x² + 4x – 36 =
4x³ –17x² + 13x – 36

Now add remainder –5:

4x³ –17x² + 13x – 36 – 5 = 4x³ –17x² + 13x – 41 ✔ Matches!

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Final Answer:
4x² – x + 9 R –5