Question

if alpha and beta are the zeros of the polynomial x square - 5 x + 4 find all the quadratic polynomial whose zeros are alpha + 1 upon beta and beta + 1 upon Alpha ​

Answer 1

Refer the attachment.

Answer 2

Answer: 4x² - 25x + 25

Step by step explanation:

If p and q are the roots of the quadratic equation, then the equation is:

x² - (p + q)x + pq = 0

Given quadratic equation:

x² - 5x + 4 = 0

Finding the zeroes,

x² - 5x + 4 = 0

» x² - 4x - x + 4 = 0

» x(x - 4) - 1(x - 4) = 0

» (x - 1) (x - 4) = 0

Thus, the zeroes of the given quadratic equation are 1 and 4.

$\tt{\alpha = 1 \: and \: \beta = 4}$

Thus,

$\tt{ \beta + \frac{1}{\alpha} = {4} + 1 = 5}$

$\tt{\alpha + \frac{1}{\beta} = 1 + \frac{1}{4} = \frac{5}{4}}$

Hence, Required possible quadratic polynomial is :

x² - (5 + 5/4)x + 5 × 5/4 = 0

x² - 25/4x + 25/4 = 0

4x² - 25x + 25 = 0

Hence,

The quadratic polynomial whose zeros are alpha + 1 upon beta and beta + 1 upon Alpha is 4x² - 25x + 25.