Question

If the polynomial 2x^3+ax^2-bx+2 is divided by (x+1) and (x-2), then the remainders are 12 and 24 respectively. Find the value of a and b.

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Answer 1

If 12 is the remainder when (x+1) is divided by 2x^3+ax^2-bx+2

So, (x+1) will be exactly divisible by 2x^3+ax^2-bx+2-12

So, x = -1

Putting the value of x

$\tt p(1) = 2(-1)^3+a(-1)^2-b(-1)+2-12 \\\\ \tt \Rightarrow -2 + a + b -10 = 0 \\\\ \tt \Rightarrow a+b = 12 \rightarrow \rightarrow \rightarrow (1)$

Now In another Case where 24 is the remainder when divided by (x-2)

So, (x-2) will be exactly divisible by

So, x = 2

Putting the value of x

$\tt p(2) = 2(2)^3+a(2)^2-b(2)+2-24 \\\\ \tt \Rightarrow 16 + 4a - 2b -22 = 0 \\\\ \tt \Rightarrow 4a - 2b = 6 \\\\ \tt \Rightarrow 2a - b = 3 \rightarrow \rightarrow \rightarrow (2)$

From eq 1 we have, a = 12 - b

Putting value of a in equation 2

$\tt 2(12-b) - b = 3 \\\\ \tt \Rightarrow 24 - 2b -b = 3 \\\\ \tt \Rightarrow -3b = -21 \\\\ \tt \Rightarrow b =7$

Putting value of b in equation 1

a + 7 = 12

=> a = 5

Answer 2

$\mathtt{\huge{\underline{\red{Question\:?}}}}$

If the polynomial 2x^3+ax^2-bx+2 is divided by (x+1) and (x-2), then the remainders are 12 and 24 respectively. Find the value of a and b.

$\mathtt{\huge{\underline{\green{Answer:-}}}}$

➡The value of a is 5 and b is 7.

$\mathtt{\huge{\underline{\purple{Solution:-}}}}$

Given :-

• 2x³+ax²-bx+2 is divided by (x+1) and (x-2)

• The remainders are 12 and 24 respectively.

To find :-

• The value of a and b.

Calculation :-

Let a polynomial P(x) = 2x³ + ax² + bx – 2

Case 1 :-

Here, 12 is the remainder.

For, (x+1) by 2x³+ax²-bx+2

So, (x+1) will be exactly divisible by

2x³+ax²-bx+2-12

(x+1)

= x = -1

Putting the value of x

p(1) = 2(-1)^3+a(-1)^2-b(-1)+2-12

➡ -2 + a + b -10 = 0

➡ a+b = 12 .........(1)

Case 2 :-

Here, 24 is the remainder.

For, (x-2) by 2x³+ax²-bx+2

So, (x-2) will be exactly divisible by

2x³+ax²-bx+2-12

(x-2)

= x = 2

Putting value of x

p(2) = 2(2)^3+a(2)^2-b(2)+2-24

➡ 16 + 4a - 2b -22 = 0

➡ 4a - 2b = 6

➡ 2a - b = 3 ............(2)

From eq 1 we have,

a+b = 12

a = 12-b....... 3

Putting 3 in equation 2,

➡ 2(12-b) - b = 3

➡ 24 - 2b -b = 3

➡ -3b = -21

➡b = -21/-3

➡ b =7

Putting value of b in equation 1

a + 7 = 12

=> a = 5

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