Question

b) When a polynomial 3x3 + 2x2 + ax + b is divided by (x + 2) leaves
remainder 4 and (x-3) leaves remainder -1. Find 'a' and 'b'.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomial is

$\sf \: {3x}^{3} + {2x}^{2} + ax + b$

Let assume that

$\sf \: f(x) = {3x}^{3} + {2x}^{2} + ax + b - - - - (1)$

It is given that, when f(x) is divided by (x + 2), it leaves the

remainder 4.

So, using Remainder Theorem, we have

$\sf \: f( - 2) = 4$

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

$\sf \: - 24+ 8 - 2 a+ b = 4$

$\sf \: - 16 - 2 a+ b = 4$

$\sf \: \sf \: \implies \: b = 2a + 20 - - - (2)$

Further, given that when f(x) is divided by (x - 3), it leaves the remainder - 1.

So, by using Remainder Theorem, we have

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

$\sf \: 81 + 18 + 3a + b = - 1$

On substituting the value of b from equation (2), we get

$\sf \: 99 + 3a + 2a + 20 = - 1$

$\sf \: 119 + 5a = - 1$

$\sf \: 5a = - 1 - 119$

$\sf \: 5a = - 120$

$\bf\implies \:a \: = \: - \: 24$

Substituting the value of a in equation (2), we get

$\sf \: b = 2( - 24) + 20$

$\sf \: b = - 48+ 20$

$\sf \:\bf\implies \: b = - 28$

$\rule{190pt}{2pt}$

Basic Concept Used :-

Remainder Theorem :- This theorem states that if a polynomial f(x) of degree greater than or equals to 1 is divided by x - a, then remainder is f(a).

$\rule{190pt}{2pt}$

Additional Information

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$