Question

if alpha and beta are thce roots of the quadratic equation 3x^2 - 7x +4=0 form a quadratic polynomial with zeroes 1) 1/alpha , 1/beta 2) alpha/beta, beta/alpha 3) alpha^2,beta^2 4) alpha^3,beta^3 5) -alpha,-beta​

Answer 1

understand the question and solve the given quadratic equation then you will get the answer

Answer 2

Answer:

(i) 4x² - 7x + 3 = 0

(ii) 12x² - 25x + 12 = 0

Step-by-step explanation:

Quadratic polynomial with roots α and β can be represented as:

(x - α)(x - β) = 0

⇒ x² - (α + β)x + αβ = 0

In the given Equation using factorization method:

3x² - 7x + 4 = 0

3x² - 3x - 4x + 4 = 0

3x(x - 1) - 4(x - 1) = 0

(3x - 4)(x - 1) = 0

∴ roots of this equation are α = $\frac{4}{3}$ and β = 1

So when the zeros are

(i)  $1/\alpha , 1/\beta$, the quadratic equation is

$x^2 - (\frac{1}{\alpha} + \frac{1}{\beta})x + \frac{1}{\alpha \beta} = 0$

⇒ $x^{2} - (\frac{3}{4} +1) x + \frac{3}{4}$

⇒ 4x² - 7x + 3 = 0

(ii) $\alpha/\beta, \beta/\alpha$ , the quadratic equation is

$x^2 - (\frac{\alpha}{\beta} + \frac{\beta}{\alpha})x + \frac{\alpha}{\beta} . \frac{\beta}{\alpha}} = 0$

⇒ $x^{2} - (\frac{3}{4} + \frac{4}{3}) x + 1 = 0$

⇒ 12x² - 25x + 12 = 0

Likewise solve for (iii), (iv) and (v)