If α and β are zeros of a polynomial 6x^2 - 5x + 1 then form a quadratic polynomial whose zeros are α and β^2. If α, β are zeros of quadratic polynomial 5x^2 + 5x + 1, find the val... If α and β are zeros of a polynomial 6x^2 - 5x + 1 then form a quadratic polynomial whose zeros are α and β^2. If α, β are zeros of quadratic polynomial 5x^2 + 5x + 1, find the value: 1. α^2 + β^2 2. α^{-1} + β^{-1}. A quadratic polynomial having zeros -√(5/2) and √(5/2) is? Read more

Steps to Solve

1. Identify the problem parts The question consists of three parts about polynomials. We need to form a polynomial with given zeros (parts 1 and 3) and evaluate certain expressions using those zeros (part 2).

2. Forming a polynomial with given zeros For part 1, we need to create a quadratic polynomial whose zeros are $\alpha$ and $\beta$. The polynomial can be formed using the equation: $$ P(x) = k(x - \alpha)(x - \beta) $$ where $k$ is a non-zero constant.

3. Calculating the value of $\alpha^2 + \beta^2$ Using the relation: $$ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta $$ we can find the required value from the quadratic polynomial in part 2, which has the form $5x^2 + 5x + 1$.

4. Solving for $\alpha + \beta$ and $\alpha\beta$ From the polynomial $5x^2 + 5x + 1$, we can identify:

• The sum of the roots ($\alpha + \beta$) is given by $-\frac{b}{a} = -\frac{5}{5} = -1$.
• The product of the roots ($\alpha\beta$) is given by $\frac{c}{a} = \frac{1}{5}$.

5. Evaluate $\alpha^{-1} + \beta^{-1}$ Using the relationship: $$ \alpha^{-1} + \beta^{-1} = \frac{\alpha + \beta}{\alpha\beta} $$ we substitute our previous results: $$ \alpha^{-1} + \beta^{-1} = \frac{-1}{\frac{1}{5}} = -5 $$

6. Identify the quadratic polynomial with the given zeros For part 3, the zeros are $\frac{-\sqrt{5}}{2}$ and $\frac{\sqrt{5}}{2}$. The corresponding polynomial can be found using the roots: $$ P(x) = x^2 - (\alpha + \beta)x + \alpha\beta $$ Calculating:

• $\alpha + \beta = 0$ (the sum of the roots)
• $\alpha\beta = -\frac{5}{4}$ (the product of the roots) Thus, the polynomial is: $$ P(x) = x^2 - 0x - \left(-\frac{5}{4}\right) = x^2 + \frac{5}{4} $$