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The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leave the remainder 19. Find the value of “a”. Also find the remainder when p(x) is divided by x + 2.

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Given polynomial = p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 

When p(x) is divided by (x + 1) leaves the remainder is 19. 

∴ p(-1) = 19 

p(-1) = (-1)4 – 2(-1)3 + 3(-1)2 – a(-1) + 3a – 7 = 19

 ⇒ 1 + 2 + 3 + a + 3a – 7 = 19 

⇒ 4a – 1 = 19 

⇒ 4a = 20 

⇒ a = 20 ÷ 4 = 5

p(x) = x4 – 2x3 + 3x2 – 5x + 8 

p(x) is divided by (x + 2), then p(-2). 

P(-2) = (-2)4 – 2(-2)3 + 3(-2)2 – 5(-2) + 8 

= 16 + 16 + 12 + 10 + 8 = 62 

Required remainder = 62.

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