Question

If f (x) = x^4 - 2x^3 + 3x^2 - ax + b is divided by x - 1 the remainders are 5 and 19 respectively, find the values of a and b

Answer 1

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Answer 2

$f(x) = {x}^{4} - 2x {}^{3} + 3x {}^{2} - ax + b$
$we \: have \: remainder \: x - 1 \\ \: \: \: then \: x - 1 = 0 \\ hence \: x = 1 \\ \: \: \: on \: putting \: x = 1 \: in \: f(x)$

$f(1) = (1) {}^{4} - 2(1) {}^{3} + 3(1) {}^{2} - a(1) + b = 5$
$1 - 2(1) + 3(1) - a(1) + b = 5$
$1 - 2 + 3 - a + b = 5$
$4 - 2 - a + b = 5$
$2 - a + b = 5$
$- a + b = 5 - 2$
$- a + b = 3$
$a - b = - 3 \: \: \: \: \: \: \: \: \: \: \: \: \: \: - (1)$


$similarly \: \\ on \: putting \: x = - 1 \: in \: f(x)$

$f(x) = {x}^{4} - 2 {x}^{3} + 3 {x}^{2} - ax + b = 19$
$( - 1) {}^{4} - 2( - 1) {}^{3} + 3( - 1) {}^{2} - a( - 1) + b = 19$
$1 - 2( - 1) + 3(1) - a( - 1) + b = 19$
$1 + 2 + 3 + a + b = 19$
$6 + a + b = 19$
$a + b = 19 - 6$
$a + b = 13 \: \: \: \: \: \: \: \: \: \: \: \: \: \: - (2)$


$now \\ adding \: equation \: (1) \: and \: (2)$


$(a + b) + (a - b) = ( - 3) + (13)$
$a + b + a - b = - 3 + 13$
$2a = 10$
$a = \frac{10}{2}$

$a = 5$

$on \: putting \: the \: value \: of \: a \: in \: \\ equation \: (2)$

$5 + b = 13$
$b = 13 - 5$
$b = 8$



$hence \\ \: \: \: the \: value \: of \: a \: is \: 5 \\ and \\ \: \: the \: value \: of \: b \: is \: 8$







$hope \: it \: helps....$