Question

when a polynomial 2x3 + 3x^2 + ax+b is divided by (x-2) leaves remainder "2" &(x+2) leaves remainder-2 find a&b​

Answer 1

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

Given polynomial is

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

Let we assume that

$\rm :\longmapsto\: f(x) = {2x}^{3} + {3x}^{2} + ax + b$

Now, it is given that, when f(x) is divided by x - 2, it leaves remainder 2.

We know,

Remainder Theorem states that if a polynomial f(x) is divided by linear polynomial x - a, the remainder is f(a)

So, using this concept, we have

$\rm :\longmapsto\:f(2) = 2$

$\rm :\longmapsto\: {2(2)}^{3} + {3(2)}^{2} + 2a + b = 2$

$\rm :\longmapsto\: 16 + 12 + 2a + b = 2$

$\rm :\longmapsto\: 28 + 2a + b = 2$

$\bf\implies \:\boxed{ \tt{ \: 2a + b = - 26 \:}} - - - (1)$

Further, it is given that, when f(x) is divided by x + 2, it leaves remainder - 2.

So, by using Remainder Theorem, we have

$\rm :\longmapsto\:f( - 2) = - 2$

$\rm :\longmapsto\: {2( - 2)}^{3} + {3( - 2)}^{2} - 2a + b = - 2$

$\rm :\longmapsto\: - 16 + 12 - 2a + b = - 2$

$\rm :\longmapsto\: - 4 - 2a + b = - 2$

$\rm :\longmapsto\: - 2a + b = - 2 + 4$

$\rm \implies\:\boxed{ \tt{ \: - 2a + b =2 \: }} - - - (2)$

On adding equation (1) and equation (2), we get

$\rm :\longmapsto\:2b = - 24$

$\bf\implies \:\boxed{ \tt{ \: b \: = \: - \: 12 \: }}$

On substituting b = - 12, in equation (2), we get

$\rm :\longmapsto\: - 2a - 12 = 2$

$\rm :\longmapsto\: - 2a = 2 + 12$

$\rm :\longmapsto\: - 2a = 14$

$\rm \implies\:\boxed{ \tt{ \: a \: = \: - \: 7 \: }}$

Hence,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{a \: = \: - \: 7} \\ \\ &\sf{b \: = \: - \: 12} \end{cases}\end{gathered}\end{gathered}}$

Additional Information :-

Factor theorem states that if a polynomial f(x) is divisible by linear polynomial x - a, then f(a) = 0.

More Identities to know :

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)

Answer 2

Answer:

$\\ \\ </p><p>\begin{gathered} \green{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{a \: = \: - \: 7} \\ \\ &\sf{b \: = \: - \: 12} \end{cases}\end{gathered}\end{gathered}}\end{gathered} \\ </p><p></p><p>$

Step-by-step explanation:

Hope it helps you