Question

On dividing p(x)=2x³-3x²+ax-3a+9 by (x+1),if the remainder is 16,then find the value of a. Then, find the remainder on dividing p(x) by x+2.​

Answer 1

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GiveN Polynomial:

• p(x)= 2x³- 3x² + ax - 3a + 9
• Remainder is 16 when divided by x + 1
• And then divided by x + 2

To FinD:

• Value of a?
• Remainder when divided by x + 2

Step-by-Step Explanation:

When we divide p(x) by x + 1, the remainder will be p(-1)

So,

⇒ p(-1) = 2(-1)³ - 3(-1)² + a(-1) - 3a + 9

⇒ p(-1) = 2(-1) - 3(1) - a - 3a + 9

⇒ p(-1) = -2 - 3 - 4a + 9

⇒ p(-1) = 4 - 4a

According to question,

⇒ p(-1) = 16

⇒ 4 - 4a = 16

⇒ -4a = 12

⇒ a = -3

Thus, the required value of a is -3 (Ans)

So, Our polynomial will be now:

⇒ p(x) = 2x³- 3x² + (-3)x - 3(-3) + 9

⇒ p(x) = 2x³ - 3x² - 3x + 18

When we divide p(x) by x + 2, the remainder will be p(-2)

⇒ p(-2) = 2(-2)³ - 3(-2)² - 3(-2) + 18

⇒ p(-2) = 2(-8) - 3(4) + 6 + 18

⇒ p(-2) = -16 - 12 + 24

⇒ p(-2) = -28 + 24

⇒ p(-2) = -4

Thus, the required remainder is -4 (Ans)

And we are done! :D

Answer 2

Answer:-

• Value of 'a' = -3

• Remainder

[when p(x) is divided by x+2] = -4

Explanation:-

Here, given that:-

=> p(x) = 2x³-3x²+ax-3a+9

=> Divided by x+1

=> Remainder is 16

According to Remainder Theorem, we know that when p(x) is divided by (x+1), the remainder is p(-1).

=> p(-1) = 16

=> 2(-1)³-3(-1)²+a(-1)-3a+9 = 16

=> -2-3-a-3a+9 = 16

=> 4-4a = 16

=> -4a = 16-4

=> -4a = 12

=> a = 12/-4

=> a = -3

Thus, the value of a is -3.

Now again according to Remainder theorem, when p(x) is divided by x+2, then the remainder will be p(-2).

=> p(-2) = 2(-2)³-3(-2)²+(-3)(-2)-3(-3)+9

=> p(-2) = 2(-8)-3(4)+6+9+9

=> p(-2) = -16-12+6+9+9

=> p(-2) = -28+24

=> p(-2) = -4

Thus, the remainder on dividing p(x) by

(x+2) is -4.