Question

if alpha and beta are the zeros of the polynomial x square - 3 x + 2 find the form the quadratic polynomial whose zeros are Alpha minus one upon alpha plus one and beta minus one upon beta plus one?​

Answer 1

Step-by-step explanation:

Given equation is

${x}^{2} - 3x + 2 = 0$

Roots are

$\alpha = 2 \: and \: \beta = 1$

Roots of the new equation are.

$\frac{ \alpha - 1}{ \alpha + 1} , \frac{ \beta - 1}{ \beta + 1}$

$= \frac{1}{3} ,0$

Hence equation is

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

Answer 2

Step-by-step explanation:

Let the Zeroes be A and B

where A = alpha and B = beta

x² - 3x + 2 = 0

ax² + bx + c = 0

where a = 1, b = -3, c = 2

Hence, by Splitting the middle term method,

Sum = b = -3

Product = a × c = 2

Thus, the factors are -1 and -2

so,

x² - x - 2x + 2 = 0

x(x - 1) -2(x - 1) = 0

(x - 1)(x - 2) = 0

so, x = 1 (= A)

x = 2 (= B)

Now, we must find a quadratic equation whose zeroes are

(A - (1/A) + 1) and (B - (1/B) + 1)

So, let the new Quadratic equation's zeroes be 'a' and 'b'

Thus,

a = (A - 1)/(A + 1)

but we know that A = 1

so,

a = (1 - 1)/(1+1) = 0/2

hence, a = 0

Now,

b = (B - 1)/(B + 1)

but we know B = 2

so,

b = (2 - 1)/(2 + 1) = 1/3

b = 1/3

Now,

we know that if the zeroes of a Quadratic equation is a and b

then, the Quadratic equation will be

x² - (a + b)x + ab = 0

so,

x² - (0 + (1/3))x + ((1/3) × 0) = 0

x² - (1/3)x = 0

Multiplying the whole equation by 3 we get

3x² - x = 0

Thus,

Alpha = 1

Beta = 2

and Quadratic equation = 3x² - x = 0

(If you haven't understood this typing, then check the writing I have given above)

Hope it helped and you understood it........All the best