The image you've uploaded explains
synthetic division, a shortcut method for dividing a polynomial by a linear factor of the form $ x - k $. Let's go through the
example shown in the image and
solve it step-by-step, then explain the process.
---
🔷 Problem:
Divide
$$
2x^3 + 6x^2 + 0x + 29 \quad \text{(note: missing } x \text{ term, so coefficient is 0)}
$$
by
$$
x + 4
$$
---
🔷 Step 1: Rewrite the divisor in the form $ x - k $
We are dividing by $ x + 4 $, which is the same as $ x - (-4) $. So:
$$
k = -4
$$
---
🔷 Step 2: Set up synthetic division
Write down the coefficients of the dividend (in order):
- Coefficients: $ 2,\ 6,\ 0,\ 29 $ (for $ x^3,\ x^2,\ x,\ \text{constant} $)
Set up the synthetic division:
```
-4 | 2 6 0 29
| -8 8 -32
------------------------
2 -2 8 -3
```
---
🔷 Step 3: Perform the steps
1. Bring down the first coefficient:
$$
2
$$
2. Multiply $ 2 \times (-4) = -8 $, write under next coefficient:
$$
6 + (-8) = -2
$$
3. Multiply $ -2 \times (-4) = 8 $, write under next:
$$
0 + 8 = 8
$$
4. Multiply $ 8 \times (-4) = -32 $, write under last:
$$
29 + (-32) = -3
$$
---
🔷 Step 4: Interpret the result
- The
last number ($ -3 $) is the
remainder.
- The other numbers ($ 2,\ -2,\ 8 $) are the
coefficients of the quotient.
Since we divided a cubic polynomial by a linear one, the quotient is quadratic:
$$
\text{Quotient} = 2x^2 - 2x + 8
$$
And the remainder is $ -3 $, so we write:
$$
\frac{2x^3 + 6x^2 + 29}{x + 4} = 2x^2 - 2x + 8 - \frac{3}{x + 4}
$$
(Note: The remainder is written over the divisor.)
---
✔ Final Answer:
$$
(2x^3 + 6x^2 + 29) \div (x + 4) = 2x^2 - 2x + 8 - \frac{3}{x + 4}
$$
---
📌 Summary of Synthetic Division Rules:
- Only works when dividing by a
linear factor $ x - k $
- Write coefficients in order (include zeros for missing terms)
- Use $ k $ from $ x - k $
- Bring down, multiply, add, repeat
- Last number is
remainder
- Other numbers give
quotient coefficients
Let me know if you'd like to try another example!
Parent Tip: Review the logic above to help your child master the concept of long and synthetic division worksheet.
