Question

when a polynomial is divided by x-1,x+1andx+2 it leaves the reminders -1,1and2 respectively.Find the reminder when it is divided by x^3+2x^2-x-2.​

Answer 1

let the polynomial be F(x)

APQ: by applying remainder theorem,

we get the following equations

$f(1) = - 1$

then, we get

$f( - 1) = 1$

and finally we get

$f( - 2) = 2$

so, we will factorise the polynomial given

${x }^{3} + 2 {x}^{2} - x - 2$

${x}^{2} (x + 2) - 1(x + 2)$

$( {x}^{2} - 1)(x + 2)$

$( x - 1)(x + 1)(x + 2)$

so, the polynomial f(x) will be such that

$f(x) = ( {x}^{3} + 2 {x}^{2} - x - 2)q(x) + r(x)$

where, q(x) is the quotient and r(x) is the remainder. since we know,

$deg(r(x)) \leqslant deg(q(x))$

where deg(r(x)) means degree of r(x).

so, let use assume,

$r(x) = a {x}^{2} + bx + c$

returning back to our initial equation we get

$f(x) = (x - 1)(x + 1)(x + 2)q(x) + (a {x}^{2} + bx + c)$

so, substituting the values woven in the question I.e. x=1,-1,2 we get the following equations

$f(1) = (1 - 1)(1 + 1)(1 + 2)q(x) + (a \times {1}^{2} + b \times 1 + c)$

which simplifies to be

$f(1) = a + b + c = - 1$

Eq... 1

similarly we get the following equations

$f( - 1) = a \times {( - 1)}^{2} + b \times ( - 1) + c$

this again simplifies to be

$f( - 1) = a - b + c = 1$

Eq... 2

again repeating we get

$f( - 2) = a \times {( - 2)}^{2} + b \times ( - 2) + c$

this again simplifies to be

$f( - 2) = 4a - 2b + c = 2$

Eq... 3

solving these equations, we will get the value of a, b and c.

adding Eq 1 and 2

$(a + b + c) + (a - b + c) \\ = ( - 1) + (1) = 0$

or

$2(a + c) = 0$

or

$c = - a$

so, then, we will be able to get the value of b in Eq 1 given

$a + b + c = - 1$

so since c= --a

$a + b + ( - a) = b = - 1$

substituting b= --1 and c= --a in Eq 3 we get

$4a - 2b + c = 2 \\ 4a - 2( - 1) + ( - a) = 2 \\ 4a + 2 - a = 2 \\ 3a = 2 - 2 = 0 \\ a = 0$

so,

c= --a=0

substituting the values of a, b and c, we will get the r(x)

$r(x) = 0 \times {x}^{2} + ( - 1) \times x + 0$

which simplifies to be

$r(x) = - x$

HOPE THIS HELPS YOU. ALTHOUGH YOU COULD HAVE IDENTIFIED THE PATTERN BY THE VALUES GIVEN IN THE QUESTION, YOU SHOUKD ALWAYS PREFER WRITING STEP-WISE IN EXAMS.

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