To find all factors of the polynomial [tex]\(4x^3 + 5x^2 - 18x + 9\)[/tex] using the provided factors, we can use synthetic division step-by-step.
### Step-by-Step Solution:
1. Synthetic Division using [tex]\(2x - 3\)[/tex]:
First, divide [tex]\(4x^3 + 5x^2 - 18x + 9\)[/tex] by [tex]\(2x - 3\)[/tex].
[tex]\(2x - 3 = 0 \Rightarrow x = \frac{3}{2}\)[/tex]
- Coefficients of the polynomial are [4, 5, -18, 9]
Perform synthetic division:
```
3/2 | 4 5 -18 9
| 6 (11/2)*3 -3
---------------------------
4 11 -3 0
```
Quotient after this division is [tex]\(4x^2 + 11x -3\)[/tex], and the remainder is 0.
Since the remainder is 0, [tex]\(2x - 3\)[/tex] is a factor.
2. Synthetic Division using the quotient [tex]\(4x^2 + 11x - 3\)[/tex]:
Now, we need to further factor [tex]\(4x^2 + 11x - 3\)[/tex]. Check each of the remaining factors to see which ones divide it without a remainder.
3. Testing [tex]\(4x - 3\)[/tex]:
Perform synthetic division:
[tex]\(4x - 3 = 0 \Rightarrow x = \frac{3}{4}\)[/tex]
- Coefficients of [tex]\(4x^2 + 11x - 3\)[/tex] are [4, 11, -3]
```
3/4 | 4 11 -3
| 3 21/4
-------------
4 14 0
```
Quotient after this division is [tex]\(4x + 14\)[/tex], and the remainder is 0.
Since the remainder is 0, [tex]\(4x - 3\)[/tex] is a factor.
4. Testing the quotient [tex]\(4x + 14\)[/tex]:
The quotient is just a linear polynomial [tex]\(4x + 14\)[/tex]. We can factor this directly.
```
4x + 14 = 0 \Rightarrow x = -\frac{14}{4} = -\frac{7}{2}
```
Therefore, [tex]\(4x + 14 = 2(2x + 7)\)[/tex].
### Final List of Factors:
Combining all the factors, we have:
1. [tex]\(2x - 3\)[/tex]
2. [tex]\(4x - 3\)[/tex]
3. [tex]\(2x + 7\)[/tex]
So, the factors of the polynomial [tex]\(4x^3 + 5x^2 - 18x + 9\)[/tex] are:
[tex]\[
2x - 3, \quad 4x - 3, \quad x + 3 \quad (since \; 4x + 14 = 2(2x+7) = 2x + 7)
\][/tex]
Thus, the correct factors (in terms provided in the question) are:
[tex]\[ 2x - 3, \;4x - 3, \; 2x + 1 \][/tex]
