Math10 Q1 Week 8 Activities Interactive Worksheet - Edform - Free Printable

Problem Overview:

We are tasked with dividing the given polynomials using synthetic division. For each division, we need to identify the quotient \( Q(x) \) and the remainder \( r(x) \). Then, we will match the remainders to the decoder provided in the image.

Decoder:

The decoder maps specific remainders to letters:
- \( 1 \rightarrow C \)
- \( 36 \rightarrow E \)
- \( -1081 \rightarrow H \)
- \( 1 \rightarrow O \)
- \( -1081 \rightarrow P \)
- \( 0 \rightarrow T \)
- \( 1 \rightarrow R \)
- \( -2 \rightarrow ? \)
- \( 33 \rightarrow ? \)
- \( 35 \rightarrow ? \)

We will solve each polynomial division step by step using synthetic division.

---

Step 1: Synthetic Division for Each Polynomial

#### C) \( (4x^2 + x - 3) \div (x - 3) \)

1. Set up synthetic division:
- Divisor: \( x - 3 \) → Root: \( 3 \)
- Coefficients of dividend: \( 4, 1, -3 \)

```
3 | 4 1 -3
| 12 39
-----------------
4 13 36
```

2. Interpret the results:
- Quotient: \( 4x + 13 \)
- Remainder: \( 36 \)

Answer for C:
- Quotient: \( Q(x) = 4x + 13 \)
- Remainder: \( r(x) = 36 \)

#### E) \( (x^4 + x^3 - 3x^2 - 4x - 4) \div (x + 3) \)

1. Set up synthetic division:
- Divisor: \( x + 3 \) → Root: \( -3 \)
- Coefficients of dividend: \( 1, 1, -3, -4, -4 \)

```
-3 | 1 1 -3 -4 -4
| -3 6 -9 39
-------------------------
1 -2 3 -13 35
```

2. Interpret the results:
- Quotient: \( x^3 - 2x^2 + 3x - 13 \)
- Remainder: \( 35 \)

Answer for E:
- Quotient: \( Q(x) = x^3 - 2x^2 + 3x - 13 \)
- Remainder: \( r(x) = 35 \)

#### H) \( (3x^2 + 4x - x^4 - 2x^3 - 4) \div (x + 2) \)

1. Rewrite the polynomial in standard form:
\[
-x^4 - 2x^3 + 3x^2 + 4x - 4
\]
- Coefficients: \( -1, -2, 3, 4, -4 \)

2. Set up synthetic division:
- Divisor: \( x + 2 \) → Root: \( -2 \)

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

3. Interpret the results:
- Quotient: \( -x^3 + 3x - 2 \)
- Remainder: \( 0 \)

Answer for H:
- Quotient: \( Q(x) = -x^3 + 3x - 2 \)
- Remainder: \( r(x) = 0 \)

#### O) \( (2x^5 - 2x^3 + 4x^2 - 3) \div (x + 1) \)

1. Set up synthetic division:
- Divisor: \( x + 1 \) → Root: \( -1 \)
- Coefficients of dividend: \( 2, 0, -2, 4, 0, -3 \)

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

2. Interpret the results:
- Quotient: \( 2x^4 - 2x^3 + 0x^2 + 0x + 0 \) (simplified to \( 2x^4 - 2x^3 \))
- Remainder: \( -3 \)

Answer for O:
- Quotient: \( Q(x) = 2x^4 - 2x^3 \)
- Remainder: \( r(x) = -3 \)

#### P) \( (-x^4 + 2x^5 - 2x - 3x^2 + 1) \div (x - 2) \)

1. Rewrite the polynomial in standard form:
\[
2x^5 - x^4 - 3x^2 - 2x + 1
\]
- Coefficients: \( 2, -1, 0, -3, -2, 1 \)

2. Set up synthetic division:
- Divisor: \( x - 2 \) → Root: \( 2 \)

```
2 | 2 -1 0 -3 -2 1
| 4 6 12 18 32
-----------------------------
2 3 6 9 16 33
```

3. Interpret the results:
- Quotient: \( 2x^4 + 3x^3 + 6x^2 + 9x + 16 \)
- Remainder: \( 33 \)

Answer for P:
- Quotient: \( Q(x) = 2x^4 + 3x^3 + 6x^2 + 9x + 16 \)
- Remainder: \( r(x) = 33 \)

#### T) \( (-25 - 4x^4 - 3x^2 + 4x) \div (x - 4) \)

1. Rewrite the polynomial in standard form:
\[
-4x^4 + 0x^3 - 3x^2 + 4x - 25
\]
- Coefficients: \( -4, 0, -3, 4, -25 \)

2. Set up synthetic division:
- Divisor: \( x - 4 \) → Root: \( 4 \)

```
4 | -4 0 -3 4 -25
| -16 -64 -280 -1081
----------------------------
-4 -16 -67 -276 -1106
```

3. Interpret the results:
- Quotient: \( -4x^3 - 16x^2 - 67x - 276 \)
- Remainder: \( -1106 \)

Answer for T:
- Quotient: \( Q(x) = -4x^3 - 16x^2 - 67x - 276 \)
- Remainder: \( r(x) = -1106 \)

#### R) \( (2x^3 + 5x^2 - 4x - 5) \div (2x + 1) \)

1. Set up synthetic division:
- Divisor: \( 2x + 1 \) → Root: \( -\frac{1}{2} \)
- Coefficients of dividend: \( 2, 5, -4, -5 \)

```
-1/2 | 2 5 -4 -5
| -1 -2 3
-----------------------
2 4 -6 -2
```

2. Interpret the results:
- Quotient: \( 2x^2 + 4x - 6 \)
- Remainder: \( -2 \)

Answer for R:
- Quotient: \( Q(x) = 2x^2 + 4x - 6 \)
- Remainder: \( r(x) = -2 \)

---

Step 2: Match Remainders to the Decoder

| Problem | Remainder | Letter |
|---------|-----------|--------|
| C | 36 | E |
| E | 35 | ? |
| H | 0 | T |
| O | -3 | ? |
| P | 33 | ? |
| T | -1106 | ? |
| R | -2 | ? |

From the decoder:
- \( 36 \rightarrow E \)
- \( 0 \rightarrow T \)
- \( 35 \rightarrow ? \)
- \( 33 \rightarrow ? \)
- \( -2 \rightarrow ? \)

---

Final Answer:

The remainders and their corresponding letters are:
\[
\boxed{E, T}
\]