Question
Use synthetic division to decompose the following polynomial into two factors knowing that (x + 2) is a factor: 10x3+11x2-27x-18
Answers
GPT-4o mini
To use synthetic division to factor the polynomial \(10x^3 + 11x^2 - 27x - 18\) given that \((x + 2)\) is a factor, we need to perform synthetic division using the root of the factor \((x + 2)\), which is \(x = -2\).
### Step 1: Set up for synthetic division
We will use the coefficients of the polynomial, which are:
- \(10\) for \(x^3\)
- \(11\) for \(x^2\)
- \(-27\) for \(x\)
- \(-18\) for the constant term
This gives us the following coefficients: \(10, 11, -27, -18\).
### Step 2: Perform synthetic division
1. Write \(-2\) to the left and the coefficients to the right:
```
-2 | 10 11 -27 -18
|
```
2. Bring down the leading coefficient \(10\):
```
-2 | 10 11 -27 -18
|
|_______
10
```
3. Multiply \(-2\) by \(10\) and write the result below the next coefficient \(11\):
```
-2 | 10 11 -27 -18
| -20
|_______
10 -9
```
4. Now, multiply \(-2\) by \(-9\) and write the result below \(-27\):
```
-2 | 10 11 -27 -18
| -20 18
|_______
10 -9 -9
```
5. Next, multiply \(-2\) by \(-9\) and write the result below \(-18\):
```
-2 | 10 11 -27 -18
| -20 18 18
|_______
10 -9 -9 0
```
Since the remainder is \(0\), the division is complete, confirming that \((x + 2)\) is indeed a factor. The resulting polynomial after synthetic division is:
\[
10x^2 - 9x - 9
\]
### Step 3: Factor the quadratic
Now we need to factor \(10x^2 - 9x - 9\). We can use the factorization method to find two numbers that multiply to \(10 \times -9 = -90\) and add up to \(-9\).
The two numbers that work are \(-15\) and \(6\). Thus, we can rewrite the quadratic:
\[
10x^2 - 15x + 6x - 9
\]
Next, we group terms:
\[
(10x^2 - 15x) + (6x - 9)
\]
Factoring out the common factors in each group:
\[
5x(2x - 3) + 3(2x - 3)
\]
Now, factor out the common factor \((2x - 3)\):
\[
(5x + 3)(2x - 3)
\]
### Step 4: Write the final factorization
Combining everything, the complete factorization of the original polynomial \(10x^3 + 11x^2 - 27x - 18\) is:
\[
(x + 2)(5x + 3)(2x - 3)
\]
Thus, the polynomial \(10x^3 + 11x^2 - 27x - 18\) is fully decomposed into factors as \((x + 2)(5x + 3)(2x - 3)\).
### Step 1: Set up for synthetic division
We will use the coefficients of the polynomial, which are:
- \(10\) for \(x^3\)
- \(11\) for \(x^2\)
- \(-27\) for \(x\)
- \(-18\) for the constant term
This gives us the following coefficients: \(10, 11, -27, -18\).
### Step 2: Perform synthetic division
1. Write \(-2\) to the left and the coefficients to the right:
```
-2 | 10 11 -27 -18
|
```
2. Bring down the leading coefficient \(10\):
```
-2 | 10 11 -27 -18
|
|_______
10
```
3. Multiply \(-2\) by \(10\) and write the result below the next coefficient \(11\):
```
-2 | 10 11 -27 -18
| -20
|_______
10 -9
```
4. Now, multiply \(-2\) by \(-9\) and write the result below \(-27\):
```
-2 | 10 11 -27 -18
| -20 18
|_______
10 -9 -9
```
5. Next, multiply \(-2\) by \(-9\) and write the result below \(-18\):
```
-2 | 10 11 -27 -18
| -20 18 18
|_______
10 -9 -9 0
```
Since the remainder is \(0\), the division is complete, confirming that \((x + 2)\) is indeed a factor. The resulting polynomial after synthetic division is:
\[
10x^2 - 9x - 9
\]
### Step 3: Factor the quadratic
Now we need to factor \(10x^2 - 9x - 9\). We can use the factorization method to find two numbers that multiply to \(10 \times -9 = -90\) and add up to \(-9\).
The two numbers that work are \(-15\) and \(6\). Thus, we can rewrite the quadratic:
\[
10x^2 - 15x + 6x - 9
\]
Next, we group terms:
\[
(10x^2 - 15x) + (6x - 9)
\]
Factoring out the common factors in each group:
\[
5x(2x - 3) + 3(2x - 3)
\]
Now, factor out the common factor \((2x - 3)\):
\[
(5x + 3)(2x - 3)
\]
### Step 4: Write the final factorization
Combining everything, the complete factorization of the original polynomial \(10x^3 + 11x^2 - 27x - 18\) is:
\[
(x + 2)(5x + 3)(2x - 3)
\]
Thus, the polynomial \(10x^3 + 11x^2 - 27x - 18\) is fully decomposed into factors as \((x + 2)(5x + 3)(2x - 3)\).