Question

if alpha and beta are zeroes of polynomial 25x2-15x+2 form a polynomial whose zeroes are alpha-beta and alpha+beta​

Answer 1

Answer:

Given:

•  alpha and beta are zeros of polynomial 25x2-15x+2

To find:

•  a polynomial whose zeroes are alpha-beta and alpha+beta​

Pre-requisite Knowledge :

if α  and β are the zeros,then,

• α + β = -b/a

• α * β = c/a

Solving Question:

We are given alpha and beta as zeros and are asked to find a polynomial with their zeros as  alpha-beta and alpha+beta​ . For that let's find the value of alpha and beta and the we can find the answer.

Solution:

25x² - 15x +2

quadratic equation: $\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

substitute the value

a = 25 ; b = -15 ; c = 2

$\dfrac{-(-15)\pm \sqrt{15^2 -4*2*25}}{2*25}\\\\\\\\\implies\dfrac{15 \pm 5}{50}$

⇒ 15+5/50 = 20/50 = 2/5

or,

15-5/50 = 10/50 = 1/5

⇒ α = 2/5

and β = 1/5

∴ zeros of the required polynomial

= alpha-beta and alpha+beta​

= 2/5 - 1/5 = 1/5

and alpha+beta​

2/5 + 1/5 = 3/5

∴ the new zeros = 3/5 and 1/5

standard form of the polynomial

ax² + bx + c = 0

α +β = -b/a

3/5 +1/5 = -b/a

4/5 = -b/a

b = -4

a = 5

α *β = c/a

3/5 * 1/5 = c/a

3/5 = c/a

c = 3

∴ 5x² -4x +3 = 0 is the required polynomial.