The polynomial P ( x )= x 3+ ax 2+ bx + c =0 has the property that the arithmetic mean of its roots, the product of its roots and the sum of its coefficients are all equal. If the y-intercept of the graph of y=P(x) is 2 , then find the value of b+P(2).

Question

The polynomial P ( x )= x 3+ ax 2+ bx + c =0 has the property that the arithmetic mean of its roots, the product of its roots and the sum of its coefficients are all equal. If the y-intercept of the graph of y=P(x) is 2 , then find the value of b+P(2). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Θ y-intercept ⇒ P(0)=c=2 ∴ text product of roots =- c =-2 text sum of roots =- a ∴ text Arithmetic mean =(- a /3)=-2 ⇒ a =6 Θ text sum of coefficient =1+ a + b + c =-2 ⇒ 1+6+ b +2=-2 ⇒ b =-11 ∴ P ( x )= x 3+6 x 2-11 x +2 ∴ P (2)=8+24-22+2=12 P (2)+ b =1

Q. The polynomial P(x)=x3+ax2+bx+c=0 has the property that the arithmetic mean of its roots, the product of its roots and the sum of its coefficients are all equal. If the y-intercept of the graph of y=P(x) is 2 , then find the value of b+P(2).

Θy-intercept ⇒P(0)=c=2 ∴ product of roots =−c=−2 sum of roots =−a ∴ Arithmetic mean =3−a​=−2⇒a=6 Θ sum of coefficient =1+a+b+c=−2 ⇒1+6+b+2=−2 ⇒b=−11 ∴P(x)=x3+6x2−11x+2 ∴P(2)=8+24−22+2=12 P(2)+b=1

Q. The polynomial $P (x) = x^{3} + a x^{2} + b x + c = 0$ has the property that the arithmetic mean of its roots, the product of its roots and the sum of its coefficients are all equal. If the $y$ -intercept of the graph of $y = P (x)$ is 2 , then find the value of $b + P (2)$ .