Practice Worksheet: Synthetic Division | Study notes Mathematics ... - Free Printable

Let's solve each problem step by step.

---

Problem 1: Write the dividend as a polynomial in standard form.

Given synthetic division setup:

```
-1 | 1 2 -1 -2
-1 -1 2
-----------------
1 1 -2 0
```

- The top row (before division) represents coefficients of the dividend.
- Since we are dividing a cubic polynomial (3rd degree, 4 coefficients), the dividend is:

x³ + 2x² - x - 2

✔ Answer: x³ + 2x² - x - 2

---

Problem 2: Write the quotient as a polynomial in standard form.

Given:

```
4 | 3 -10 0 -5
12 8 32
-----------------
3 2 8 27
```

- The bottom row (excluding remainder) gives the quotient coefficients.
- Since we started with a cubic (degree 3), quotient is quadratic (degree 2): 3x² + 2x + 8
- Remainder is 27 (not part of quotient).

✔ Answer: 3x² + 2x + 8

---

Problem 3: Write the dividend as a product of two factors.

Given:

```
-2 | 1 2 1 5 6
-2 0 -2 -6
-------------------
1 0 1 3 0
```

- Remainder is 0 → so (x + 2) is a factor.
- Quotient coefficients: 1, 0, 1, 3 → corresponds to x³ + 0x² + x + 3 = x³ + x + 3
- So, dividend = (x + 2)(x³ + x + 3)

✔ Answer: (x + 2)(x³ + x + 3)

---

Problem 4: Fill in missing values to complete synthetic division.

Given:

```
□ | 2 1 □ 10
-4 6 □
----------------
□ -3 5 □
```

We’ll fill step by step.

Let’s denote the divisor as c.

First number in quotient row is 2 (same as first coefficient).

Then: c * 2 = -4 → so c = -2

Now, proceed:

- Bring down 2.
- Multiply 2 * (-2) = -4 → write under next coefficient (1).
- Add: 1 + (-4) = -3 → matches given.
- Multiply -3 * (-2) = +6 → write under next coefficient (which is blank).
- Add: blank + 6 = 5 → so blank = -1
- Multiply 5 * (-2) = -10 → write under 10.
- Add: 10 + (-10) = 0

So filled table:

```
-2 | 2 1 -1 10
-4 6 -10
----------------
2 -3 5 0
```

✔ Missing values:
- Divisor: -2
- Third coefficient of dividend: -1
- Last multiplier: -10
- Remainder: 0

---

Problem 5: Fill in missing values to complete synthetic division.

Given:

```
□ | □ -4 1 □
3 □ □
----------------
□ -1 -2 0
```

Let’s denote divisor as c.

First, bottom row starts with some value — let’s call it a (first coefficient of quotient).

We know:

- First coefficient of dividend is a (since it’s brought down).
- Then: c * a = 3 → equation (1)
- Next: -4 + (c * a) = -1 → but c*a = 3, so -4 + 3 = -1 → ✔ checks out.
- Next: c * (-1) = ? → this goes under the 1.
Let’s call that value b → so b = c*(-1) = -c
- Then: 1 + b = -2 → 1 + (-c) = -2 → -c = -3 → c = 3

From equation (1): c * a = 3 → 3a = 3 → a = 1

Now continue:

- c = 3
- First coefficient of dividend: 1
- Multiply 3 * (-1) = -3 → write under 1 → 1 + (-3) = -2 → ✔
- Multiply 3 * (-2) = -6 → write under last coefficient (blank)
- Add: blank + (-6) = 0 → so blank = 6

Final table:

```
3 | 1 -4 1 6
3 -3 -6
----------------
1 -1 -2 0
```

✔ Missing values:
- Divisor: 3
- First coefficient of dividend: 1
- Second multiplier: -3
- Third multiplier: -6
- Last coefficient of dividend: 6

---

Problem 6: (2x³ + 4x² - 3x - 6) ÷ (x + 3)

Use synthetic division with root = -3

Coefficients: 2, 4, -3, -6

```
-3 | 2 4 -3 -6
-6 6 -9
--------------------
2 -2 3 -15
```

Quotient: 2x² - 2x + 3, Remainder: -15

✔ Answer: 2x² - 2x + 3 - 15/(x + 3)
*(But since question says “find the quotient”, we give polynomial quotient only:)*
➡️ 2x² - 2x + 3

---

Problem 7: (x³ - 6x² + 9) ÷ (x - 4)

Note: Missing x term → coefficient of x is 0.

Coefficients: 1, -6, 0, 9

Divide by x - 4 → use root = 4

```
4 | 1 -6 0 9
4 -8 -32
-----------------
1 -2 -8 -23
```

Quotient: x² - 2x - 8, Remainder: -23

✔ Answer: x² - 2x - 8

---

Problem 8: (4x³ + 27x² + 3x + 64) ÷ (x + 7)

Root = -7

Coefficients: 4, 27, 3, 64

```
-7 | 4 27 3 64
-28 7 -70
------------------
4 -1 10 -6
```

Quotient: 4x² - x + 10, Remainder: -6

✔ Answer: 4x² - x + 10

---

Problem 9: (x⁴ - 16x² - 40x - 25) ÷ (x - 5)

Note: Missing x³ term → coefficient = 0

Coefficients: 1, 0, -16, -40, -25

Root = 5

```
5 | 1 0 -16 -40 -25
5 25 45 25
------------------------
1 5 9 5 0
```

Remainder = 0 → so it divides evenly.

Quotient: x³ + 5x² + 9x + 5

✔ Answer: x³ + 5x² + 9x + 5

---

## ✔ Final Answers Summary:

1. x³ + 2x² - x - 2
2. 3x² + 2x + 8
3. (x + 2)(x³ + x + 3)
4. Divisor: -2, dividend coeffs: 2, 1, -1, 10, multipliers: -4, 6, -10, remainder: 0
5. Divisor: 3, dividend coeffs: 1, -4, 1, 6, multipliers: 3, -3, -6, remainder: 0
6. 2x² - 2x + 3
7. x² - 2x - 8
8. 4x² - x + 10
9. x³ + 5x² + 9x + 5

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