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. $\alpha^4 + \beta^4 = \frac{23}{9}$

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.