This question is taken from polynomials in which we have given a cubic equation p(x)=(3x³-10x²-2x+10) and we have to show that 5,-2,1/3 are its zeroes and we also have to verify the relationship between the zeroes and coefficient.

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 2B, question no 2

Let f(x)=3x³−10x²−27x+10

5,−2 and 1/3 are the zeroes of the polynomial ( given )

Therefore,

f(5)=3(5)³−10(5)²−27(5)+10

=3×125−250−135+10

=0

f(−2)=3(−2)³−10(−2)²−27(−2)+10

−24−40+54+10

=0

f(1/3)=3(1/3)³−10(1/3)²−27(1/)³+10

=1/8−10/9−9+10

=0

Verify relations :

General form of cubic equation :ax³+bx²+cx+d

now ,

Consider α=5,β=−2 and y=31

α+β+y=5−2+1/3=10/3=−b/a

αβ+βy+αy=5(−2)+(−2)(1/3)+(5×1/3)=−10+(−2)/3+5/3=−9=c/a

and αβy=5(−2)(1/3)=−10/3=−d/a