Question

find valu of m if the equation is divisible by 2y-1​

Answer 1

ANSWER:

Given:

• p(y) = 2y^3+my^2+11y+m+3
• p(y) is divisible by 2y - 1.

To Find

• value of m

Solution:

We are given that,

$\implies p(y) = 2y^3+my^2+11y+m+3$

We are also given that, p(y) is divisible by 2y - 1.

So, 2y - 1 is a factor of p(y).

$\implies 2y-1=0$

$\implies y=\dfrac{1}{2}$

So, 1/2 is a zero of p(y).

$\implies p(1/2) =0$

$\implies p(y) = 2y^3+my^2+11y+m+3$

$\implies p\left(\dfrac{1}{2}\right) = 2\left(\dfrac{1}{2}\right)^3+m\left(\dfrac{1}{2}\right)^2+11\left(\dfrac{1}{2}\right)+m+3$

So,

$\implies p\left(\dfrac{1}{2}\right) = 2\left(\dfrac{1}{8}\right)+m\left(\dfrac{1}{4}\right)+11\left(\dfrac{1}{2}\right)+m+3$

$\implies p\left(\dfrac{1}{2}\right) = \dfrac{1}{4}+\dfrac{m}{4}+\dfrac{11}{2}+m+3$

$\implies p\left(\dfrac{1}{2}\right) =\dfrac{1}{4}+\dfrac{m}{4}+\dfrac{22}{4}+\dfrac{4m}{4}+\dfrac{12}{4}$

$\implies 0 =\dfrac{1+m+22+4m+12}{4}$

$\implies 0 =\dfrac{5m+35}{4}$

$\implies 5(m+7)=4\times0$

$\implies 5(m+7)=0$

$\implies m+7=0$

Hence,

$\implies\bf m=-7$

Therefore, the value of m is -7.