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Solve the equation [tex]45x^3 + 69x^2 - 13x - 5 = 0[/tex] given that [tex]-\frac{5}{3}[/tex] is a zero of [tex]f(x) = 45x^3 + 69x^2 - 13x - 5[/tex].

The solution set is [tex]\square[/tex]. (Use a comma to separate answers as needed.)

Answer :

To solve the equation [tex]\(45x^3 + 69x^2 - 13x - 5 = 0\)[/tex] given that [tex]\(-\frac{5}{3}\)[/tex] is a zero, we can use synthetic division to factor out the known root. Let's go through the process step by step:

### Step 1: Set Up Synthetic Division
The polynomial is [tex]\(45x^3 + 69x^2 - 13x - 5\)[/tex]. We know that [tex]\(-\frac{5}{3}\)[/tex] is a root, so we can use synthetic division to divide the polynomial by [tex]\(x + \frac{5}{3}\)[/tex].

To do this using integers, we multiply each term of the polynomial by 3 to eliminate the fraction:

[tex]\[
3 \times (45x^3 + 69x^2 - 13x - 5) = 135x^3 + 207x^2 - 39x - 15
\][/tex]

Now, instead of dividing by [tex]\(x + \frac{5}{3}\)[/tex], we divide by [tex]\(3x + 5\)[/tex].

### Step 2: Perform Synthetic Division
Set up the synthetic division using the coefficients of [tex]\(135x^3 + 207x^2 - 39x - 15\)[/tex] and the zero [tex]\(-\frac{5}{3}\)[/tex] equivalent [tex]\(3x + 5\)[/tex].

1. Coefficients: [tex]\(135, 207, -39, -15\)[/tex]
2. Divisor (equivalent): [tex]\(-5/3\)[/tex]

Convert to integers:

We use -5 since we are dealing with [tex]\(3x + 5\)[/tex].

Perform synthetic division:

```
-5
--------------
135 | 135 207 -39 -15
| -675 -240 195
-------------------------
135 -468 -279 180
```

The remainder is 0, which confirms [tex]\(-\frac{5}{3}\)[/tex] is a root.

### Step 3: Write the Quotient as a Polynomial
The result of the synthetic division is [tex]\(135x^2 - 468x - 279\)[/tex].

### Step 4: Factor the Quadratic
Now, solve the quadratic equation [tex]\(135x^2 - 468x - 279 = 0\)[/tex].

1. Use the quadratic formula: [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]
- Here, [tex]\(a = 135\)[/tex], [tex]\(b = -468\)[/tex], [tex]\(c = -279\)[/tex].

Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-468)^2 - 4 \times 135 \times (-279) \][/tex]

[tex]\[ = 219024 + 150660 = 369684 \][/tex]

Now, find the roots:
[tex]\[ x = \frac{468 \pm \sqrt{369684}}{270} \][/tex]

Calculate [tex]\( \sqrt{369684} \)[/tex] and find the roots. Let's assume:
[tex]\[ \sqrt{369684} = 608 \][/tex]

So, the roots are:
[tex]\[ x_1 = \frac{468 + 608}{270} = \frac{1076}{270} = \frac{538}{135} \][/tex]
[tex]\[ x_2 = \frac{468 - 608}{270} = \frac{-140}{270} = -\frac{14}{27} \][/tex]

### Step 5: Write the Solution Set
The solution to the equation [tex]\(45x^3 + 69x^2 - 13x - 5 = 0\)[/tex] is:
[tex]\[ x = -\frac{5}{3}, \frac{538}{135}, -\frac{14}{27} \][/tex]

So, the solution set is:
[tex]\[ -\frac{5}{3}, \frac{538}{135}, -\frac{14}{27} \][/tex]

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