Question

if α and β are the zeroes of the polynomial ax^2+bx+c then form the polynomial whose,zeroes are 1+α and 1+β


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Answer 1

Here is the stepwise solution. I hope u find it useful
Given:
α and β are zeroes of ax² + bx + c
To Find:
Polynomial with zeroes 1 + α and 1 + β

By the quadratic formula:
 
x = (-b +- √(b² - 4ac)/2a
 Therefore, the zeroes are 
 α = ( -b + √b² - 4ac)/2a and β = (-b - √b² - 4ac)/2a

new zeroes are α' = α+1 and β' = β +1

α' + β' = -b'/a'
( -b + √b² - 4ac) + 2a + ( -b - √b² - 4ac) + 2a/2a = b'/a'
(4a-2b)/2a = b'/a'
2a-b/a = -b'/a'

α'×β' = c'/a'
{[( -b + √b² - 4ac) +2a/2a  ] × [( -b - √b² - 4ac) + 2a/2a]}/4a² = c'/a'
{b² - b² + 4ac + 2a( -b - √b² - 4ac) + 2a( -b + √b² - 4ac) + 4a²}/4a² = c'/a'
{4ac - 2ab -2a√b²-4ac + 2a√b² - 4ac -2ab + 4a² = c'/a}/4a² = c'/a'
a + c - b/a = c'/a'
∴ The polynomial is a'x² + b'x +c'
⇒ the polynomial is ax² + (2a-b)x + a + c - b