Answer :
To solve the equation [tex]\(36x^3 + 45x^2 - 7x - 4 = 0\)[/tex] given that [tex]\(-\frac{4}{3}\)[/tex] is a zero, we can follow these steps:
1. Recognize the Factor:
Since [tex]\(-\frac{4}{3}\)[/tex] is a zero of the polynomial, it means that [tex]\((x + \frac{4}{3})\)[/tex] is a factor of the polynomial.
2. Perform Polynomial Division:
We will perform synthetic division to divide the polynomial by [tex]\((x + \frac{4}{3})\)[/tex]. This will help us simplify the polynomial and find other possible roots.
3. Synthetic Division:
The coefficients of the polynomial are [tex]\(36, 45, -7, -4\)[/tex]. We perform synthetic division using [tex]\(-\frac{4}{3}\)[/tex]:
- Start with the first coefficient, [tex]\(36\)[/tex].
- Multiply it by [tex]\(-\frac{4}{3}\)[/tex] and add it to the next coefficient:
- [tex]\(36 + 0 = 36\)[/tex],
- [tex]\(-\frac{4}{3} \times 36 + 45 = -48 + 45 = -3\)[/tex],
- [tex]\(-\frac{4}{3} \times (-3) + (-7) = 4 + (-7) = -3\)[/tex],
- [tex]\(-\frac{4}{3} \times (-3) + (-4) = 4 - 4 = 0\)[/tex].
As a result, we have a quadratic polynomial: [tex]\(36x^2 - 3 = 0\)[/tex].
4. Solve the Quadratic Equation:
The quadratic equation is [tex]\(36x^2 - 3 = 0\)[/tex].
- Simplify by dividing each term by 3:
[tex]\[ 12x^2 - 1 = 0 \][/tex]
- Add 1 to both sides:
[tex]\[ 12x^2 = 1 \][/tex]
- Divide by 12:
[tex]\[ x^2 = \frac{1}{12} \][/tex]
- Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{1}{12}} \][/tex]
Simplifying, the roots are:
[tex]\[ x = \frac{1}{\sqrt{12}} = \pm \frac{1}{2\sqrt{3}} \][/tex]
[tex]\[ x = \pm \frac{\sqrt{3}}{6} \][/tex]
5. Write the Solution Set:
The solution set for the equation is:
[tex]\[ x = -\frac{4}{3}, \ \frac{\sqrt{3}}{6}, \ -\frac{\sqrt{3}}{6}. \][/tex]
So, the complete solution to the equation [tex]\(36x^3 + 45x^2 - 7x - 4 = 0\)[/tex] with [tex]\(-\frac{4}{3}\)[/tex] as a zero is:
[tex]\[ -\frac{4}{3}, \ \frac{\sqrt{3}}{6}, \ -\frac{\sqrt{3}}{6}. \][/tex]
1. Recognize the Factor:
Since [tex]\(-\frac{4}{3}\)[/tex] is a zero of the polynomial, it means that [tex]\((x + \frac{4}{3})\)[/tex] is a factor of the polynomial.
2. Perform Polynomial Division:
We will perform synthetic division to divide the polynomial by [tex]\((x + \frac{4}{3})\)[/tex]. This will help us simplify the polynomial and find other possible roots.
3. Synthetic Division:
The coefficients of the polynomial are [tex]\(36, 45, -7, -4\)[/tex]. We perform synthetic division using [tex]\(-\frac{4}{3}\)[/tex]:
- Start with the first coefficient, [tex]\(36\)[/tex].
- Multiply it by [tex]\(-\frac{4}{3}\)[/tex] and add it to the next coefficient:
- [tex]\(36 + 0 = 36\)[/tex],
- [tex]\(-\frac{4}{3} \times 36 + 45 = -48 + 45 = -3\)[/tex],
- [tex]\(-\frac{4}{3} \times (-3) + (-7) = 4 + (-7) = -3\)[/tex],
- [tex]\(-\frac{4}{3} \times (-3) + (-4) = 4 - 4 = 0\)[/tex].
As a result, we have a quadratic polynomial: [tex]\(36x^2 - 3 = 0\)[/tex].
4. Solve the Quadratic Equation:
The quadratic equation is [tex]\(36x^2 - 3 = 0\)[/tex].
- Simplify by dividing each term by 3:
[tex]\[ 12x^2 - 1 = 0 \][/tex]
- Add 1 to both sides:
[tex]\[ 12x^2 = 1 \][/tex]
- Divide by 12:
[tex]\[ x^2 = \frac{1}{12} \][/tex]
- Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{1}{12}} \][/tex]
Simplifying, the roots are:
[tex]\[ x = \frac{1}{\sqrt{12}} = \pm \frac{1}{2\sqrt{3}} \][/tex]
[tex]\[ x = \pm \frac{\sqrt{3}}{6} \][/tex]
5. Write the Solution Set:
The solution set for the equation is:
[tex]\[ x = -\frac{4}{3}, \ \frac{\sqrt{3}}{6}, \ -\frac{\sqrt{3}}{6}. \][/tex]
So, the complete solution to the equation [tex]\(36x^3 + 45x^2 - 7x - 4 = 0\)[/tex] with [tex]\(-\frac{4}{3}\)[/tex] as a zero is:
[tex]\[ -\frac{4}{3}, \ \frac{\sqrt{3}}{6}, \ -\frac{\sqrt{3}}{6}. \][/tex]
