Question

The value of m, if 2y^3 + my^2 + 11y + m + 3 is
exactly divisible by 2y - 1 is
and correct option is -7.
please do it.​

Answer 1

Answer:

-7

Step-by-step explanation:

put value of y=1/2 in cubic polynomial and solve from m since 2y-1 is a root equation of this equation so it have to must satisfy it

Answer 2

Given that:

• $\sf p(y) = 2y^3+my^2+11y+m+3$

• Where p(y) is divisible by 2y - 1. Therefore, 2y - 1 is a factor of p(y).

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$\green {\sf:\implies 2y-1=0}$

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

• So, 1/2 is a zero of p(y).

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$\qquad\bigstar\:{\underline{\pmb{\frak{\pink{According~to~the~Question~:}}}}}$

$\\ \qquad\dashrightarrow\sf p(1/2) =0$

$\\ \qquad\dashrightarrow\sf p(y) = 2y^3+my^2+11y+m+3$

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

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

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

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

$\\ \qquad\dashrightarrow \sf 0 =\dfrac{1+m+22+4m+12}{4}$

$\\ \qquad\dashrightarrow \sf 0 =\dfrac{5m+35}{4}$

$\\ \qquad\dashrightarrow\sf 5(m+7)=4\times0$

$\\ \qquad\dashrightarrow \sf m+7=0$

$\\ \qquad\dashrightarrow{\underline{\boxed{\pmb{\frak{m = -7}}}}}\:\bigstar$

$\therefore\:{\underline{\sf{The \: value \:of\:m\:is\:{- 7 }}}}.$

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