Question

Given that the roots of x^3 + 3x + 2 = 0 are a, B and y
find a^2 + B^2 + y^2 without calculating the values of the root.

Answer 1

Answer:

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

$x = - 2 \: and \: y = - 1$

♠ Hope this may help you ♠

Answer 2

→ Hey user ,

→ Given Question:-

⇒ Given that the roots of x³ + 3x  + 2 = 0

  are α, β, and y. find α² + β² + y² without calculating

  the values of the root.

→ To find:-

⇒ find α² + β² + y² without calculating the values

   of the root.

→ Solution:-

→ ( x - α ) ( x - β ) ( x - y ) = 0

→ x³ - ( α + β + y ) x² + ( αβ + βy + yα ) x - αβy = 0

→ α + β + y = 0

→ αβ + βy + yα = 3

→ αβy = -2

→ We want ,

→ α² + β² + y²           [ α + β + y = 0]

→                               [ αβ + βy + yα = 3]

→                               [ αβy = -2]

→ ( α + β + y)² = ( α + β + y ) ( α + β + y)      

→                     = α² + β² + y² + 2(αβ + βy + yα)

→ α² + β² + y² = ( α + β + y)² - 2( αβ + βy + yα )

→                     = 0 - 2 ( 3 )

→                     = -6

--------------------------------------------------------------------------------