Question

If α and β are zeros of the poly x²+4x +1 ,Without finding zeros Find α and β

Whoever answers first, I'll mark them as brainiest... ​

Answer 1

If α & β are zeroes of f(x) =x²-x-2

then

α + β = -b/a = 1

α β = c/a = -2

let α’ =2α+1 & β’ = 2β+1

α’ + β’ = 2α+1 + 2β+1

=2α+2β+2 =2(α+β)+2 =2(1)+2=4

α’ β’=(2α+1)(2β+1) = 4α β+2α+ 2β+1 = 4(-2)+2(1)+1=-8+2+1=-5

Polynomial having α’ & β’ as zeroes is given by

k (x²-(α’ + β’)x +α’ β’)

= k (x² - 4x -5) Answer

By giving different values to k, there can be infinite polynomials

Answer 2

Given,

• x² + 4x + 1

To find,

• Find α and β

Solution,

Let P(x) = x² - 4x - 8

Zero of polynomial is the value of x where P(x) =0

Putting P(x) = 0

• x² - 4x - 8 = 0

We find roots using splitting the middle term method

Splitting the middle term method We need to find two numbers where

• Splitting the middle term method We need to find two numbers whereSum = - 4

• Product = -8 × 1= - 8

x² - 4x + 2x - 8 = 0

x - 8 = 0x(x - 2) + 4(x - 4 ) = 0

( x + 2 ) ( x - 4 ) = 0

4 ) = 0So x = 2, - 4

Therefore, α=- 2 & β = - 4 are the zeroes of the polynomial

P(x) = x² - 4x - 8

= x² - 4x - 8

Comparing with a ² + bx + c

So, a = 1, b = - 4, C = - 8

We verify

• Sum of zeroes = coefficient of x / coefficient of x²

I.e α + β = - 1/α

LHS = RHS

= α + β = - b/a

= α + β = - b/a

= - 2 + 4 = - 4/1

= 4 = 4

• Product of zeroes = coefficient of x / coefficient of x²

I.e α × β = c/a

LHS = RHS

α β = c/a

(-2)(4)= - 8/1

-8= - 8

Since, RHS = LHS