https://file.aiquickdraw.com/mathbot/file/1728312874945euib8qom.jpg

Let's go through the problem step by step.

### Given:
The quadratic equation is:
\[
3x^2 + 3x - 1 = 0
\]
The roots of this equation are \( \alpha \) and \( \beta \). We are tasked with evaluating expressions like \( \alpha^2 + \beta^2 \), \( \alpha^3 + \beta^3 \), and \( \alpha^4 + \beta^4 \).

### Step 1: Use Vieta’s Formulas
Vieta's formulas give relationships between the coefficients of the quadratic equation and the sums and products of its roots. For the equation \( ax^2 + bx + c = 0 \), the relationships are:
\[
\alpha + \beta = -\frac{b}{a}, \quad \alpha \beta = \frac{c}{a}
\]
Here, \( a = 3 \), \( b = 3 \), and \( c = -1 \).

- \( \alpha + \beta = -\frac{3}{3} = -1 \)
- \( \alpha \beta = \frac{-1}{3} = -\frac{1}{3} \)

### Step 2: Find \( \alpha^2 + \beta^2 \)
Using the identity:
\[
\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta
\]
Substitute the known values:
\[
\alpha^2 + \beta^2 = (-1)^2 - 2 \times \left(-\frac{1}{3}\right) = 1 + \frac{2}{3} = \frac{5}{3}
\]

### Step 3: Find \( \alpha^3 + \beta^3 \)
Using the hint provided on the page:
\[
\alpha^3 + \beta^3 = (\alpha + \beta)\left(\alpha^2 + \beta^2\right) - 3\alpha\beta(\alpha + \beta)
\]
Substitute the known values:
\[
\alpha^3 + \beta^3 = (-1) \times \frac{5}{3} - 3 \times \left(-\frac{1}{3}\right) \times (-1)
\]
\[
\alpha^3 + \beta^3 = -\frac{5}{3} - 1 = -\frac{8}{3}
\]

### Step 4: Find \( \alpha^4 + \beta^4 \)
Using the second hint:
\[
\alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha \beta)^2
\]
Substitute the known values:
\[
\alpha^4 + \beta^4 = \left(\frac{5}{3}\right)^2 - 2 \times \left(-\frac{1}{3}\right)^2
\]
\[
\alpha^4 + \beta^4 = \frac{25}{9} - 2 \times \frac{1}{9} = \frac{25}{9} - \frac{2}{9} = \frac{23}{9}
\]

### Final Answers:
1. \( \alpha^2 + \beta^2 = \frac{5}{3} \)
2. \( \alpha^3 + \beta^3 = -\frac{8}{3} \)
3. \( \alpha^4 + \beta^4 = \frac{23}{9} \)

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Would you like further details or have any questions?

### Related Questions:
1. How do you derive Vieta's formulas for higher-degree polynomials?
2. Can we generalize the identity for \( \alpha^n + \beta^n \) for any \( n \)?
3. How do the roots of a cubic equation relate to its coefficients?
4. What is the geometric significance of symmetric polynomials in the context of root-coefficient relations?
5. How does the discriminant of a quadratic equation help in analyzing its roots?

### Tip:
Always remember to use identities like \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \) to simplify expressions involving higher powers of the roots.

If the roots of 3x^2 + 3x - 1 = 0 are α and β, evaluate α + β, αβ, α^2 + β^2, α^3 + β^3, and α^4 + β^4.

Explore how to evaluate expressions involving the roots of the quadratic equation 3x^2 + 3x - 1 = 0 using Vieta’s formulas and polynomial identities. Learn how to calculate α + β, αβ, and powers of the roots with step-by-step solutions.

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