Question

Plz answer this question ​

Answer 1

SOLUTION ☺️

In the given statement, A.P. means Arithmetic Progression

f(x)= 2x^3-15x^2+37x-30

let the zeroes of polynomial be (a-b), a and (a+b)

therefore,

f(x)= 2x^3-15x^2+37x-30

=)[x-(a-b)][(x-a)(x-(a+b))]

=)(x-a)[x-(a-b)][x-(a+b)]

=)(x-a)[x^2-x(a+b)-x(a-b)-(a^2-b^2)]

=)(x-a)[x^2-ax-bx-ax+bx-a^2+b^2]

=)(x-a)[x^2-2ax-a^2+b^2]

=)x^3-2ax^2-a^2x+b^2x-ax^2+2a^2x+a^3-ab^2

=)x^3-3ax^2+a^2x+b^2x+a^3-ab^2

=)x^3-3ax^2+(a^2+b^2)x+ a^3-ab^2= 2x^3-15x^2+37x-30

Now comparing the coefficient of

x^3, x^2, x & constant terms, we get

=) -3a= -15

=) a= -15/-3

=) a= 5

And

5^2+ b^2= 37

=) 25+ b^2= 37

=) b^2= 37- 25

=) b^2= 12

=) b= √12

=) b= 2√3

Thus, the zeros of the polynomial are-

(5-2√3), 5 and (5+2√3)

HOPE it helps ✔️