Question

if x+1 is a factor of 2x^3 +ax^2+2bx+1 and 2a-3b=4,then find value of a and b

pls explain ​

Answer 1

$\large\bf{\underline{\underline{Given\: that }}}$

$\color{red}\boxed{\sf{(x+1)\: is \:a\: factor\: of\: p(x)}}$

$\large\bf{\underline{\underline{Therefore,}}}$

$\color{orange}\boxed{\sf{-1 \:is \:a \:zero \:of \:given\: p(x)}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\pink{\bigstar}★ \large\underline{\boxed{\bf\green{p(x) = 2x3 +ax2 +2bx + 1}}}$

$\textbf{Substituting the value of -1 in the given p(x)}$

$\large\bf{\underline{\underline{We\: get }}}$

$\sf{p(x) = 2 * (-1)3 + a * (-1)2 + 2 * b * (-1) + 1}$

$\sf{-2 + a -2b + 1}$

$\sf{-1 + a - 2b}$

$\textbf\color{red}{Or}$

$\sf{a - 2b = 1}$

$\sf{2a - 3b = 4}$

$\sf{a - 2b = 1 ...(1)}$

$\sf{2a - 3b = 4 ...(2)}$

$\sf{(1) 2 = 2a - 4b = 2 ...(3)}$

$\sf{(2) 1 = 2a - 3b = 4 ...(4)}$

$\sf{(4) - (3) = [ 2a - 2a ] + [ -3b - (-4b) ] = 4-2}$

$\sf{-3b + 4b = 2}$

$\large\bf{\underline{\underline{therefore}}}$

$\color{green}\boxed{\sf{b=2}}$

$\textbf{substituting the value of b in (3)}$

$\sf{2a - 4b = 2}$

$\sf{2a - (4*2) = 2}$

$\sf{2a - 8 = 2}$

$\sf{2a = 2 + 8}$

$\sf{2a = 10}$

$\sf{a = 10/2}$

$\tt\color{orchid}{So}$

$\textbf{We get the values of a and b}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\pink{\bigstar}★ \large\underline{\boxed{\bf\green{that\: is: \:a = 5 \:and \:b=2}}}$

Thankyou :)

You can also refer to the attachment for better understanding :)