Synthetic Division of Polynomials | Method & Examples - Lesson ... - Free Printable

You've uploaded an image showing a polynomial division problem using synthetic division. Let's go through the problem step by step and explain the solution.

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🔹 Problem:

Divide the polynomial:

$$
x^3 - 7x^2 + 13x + 3 \quad \text{by} \quad (x - 2)
$$

This is written as:

$$
(x^3 - 7x^2 + 13x + 3) \div (x - 2)
$$

We are using synthetic division, which is a shortcut method for dividing polynomials by linear factors of the form $ (x - c) $. Here, $ c = 2 $, since we're dividing by $ (x - 2) $.

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🔹 Step-by-step Synthetic Division:

We set up synthetic division with:

- The root from $ (x - 2) $: $ \boxed{2} $
- Coefficients of the polynomial: $ 1, -7, 13, 3 $

```
| 1 -7 13 3
2 | 2 -10 6
---------------------
1 -5 3 9
```

Let’s walk through it:

1. Bring down the first coefficient:
$ 1 $ → becomes the first number in the bottom row.

2. Multiply $ 1 \times 2 = 2 $, write under $ -7 $.
Add: $ -7 + 2 = -5 $

3. Multiply $ -5 \times 2 = -10 $, write under $ 13 $.
Add: $ 13 + (-10) = 3 $

4. Multiply $ 3 \times 2 = 6 $, write under $ 3 $.
Add: $ 3 + 6 = 9 $

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🔹 Interpretation of Results:

The bottom row gives us:
$$
1 \quad -5 \quad 3 \quad 9
$$

- The first three numbers are the coefficients of the quotient.
- The last number is the remainder.

So:

- Quotient: $ x^2 - 5x + 3 $
- Remainder: $ 9 $

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🔹 Final Answer:

$$
\frac{x^3 - 7x^2 + 13x + 3}{x - 2} = x^2 - 5x + 3 + \frac{9}{x - 2}
$$

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✔ Summary:

- Quotient: $ x^2 - 5x + 3 $
- Remainder: $ 9 $
- Division result:
$$
x^2 - 5x + 3 + \frac{9}{x - 2}
$$

✔ This matches the synthetic division shown in your image.

Let me know if you'd like to verify this using long division or check the remainder theorem!