Question

if the zeros of the polynomial 2 x cube -15 X square + 37 x - 30 are a - b , a,a+ b find all the zeros​

Answer 1

f ( x ) = 2x³ - 15x² + 37x - 30

Zeroes = ( a - b ), ( a ) and ( a + b )

On comparing the polynomial with ax³ + bx² + cx + d, we get

a = 2, b = - 15, c = 37, d = - 30

Let the zeroes be α, β and γ.

•°• α = ( a - b )

β = a

γ = ( a + b )

Now,

Sum of zeroes = α + β + γ = - b/a

a - b + a + a + b = - ( - 15 )/2

3a = 15/2

a = 5/2

Product of zeroes = αβγ = - d/a

( a - b )( a )( a + b ) = - ( - 30 )/2

Putting the value of a, we get

[ (5/2) - b ] (5/2) [ (5/2) + b ] = 15

${\boxed{\sf{(a - b)(a + b) = a^2 - b^2}}}$

[ (5/2)² - b² ] (5/2) = 15

[ (25/4) - b² ] = 6

b² = (25/4) - 6

b² = (25 - 24)/6

b² = 1/4

b = √(1/4)

b = ± 1/2

Now,

$\sf{\underline{Case \ I \ :}}$ When a = 5/2 and b = 1/2

α = a - b = 5/2 - 1/2 = 2

β = a = 5/2

γ = a + b = 5/2 + 1/2 = 3

$\sf{\underline{Case \ II \ :}}$ When a = 5/2 and b = - 1/2

α = a - b = 5/2 - ( - 1/2 ) = 5/2 + 1/2 = 3

β = a = 5/2

γ = a + b = 5/2 - (1/2) = 5/2 - 1/2 = 2

Hence,

the zeroes are 2, 3 and 5/2.