The polynomial p{x) = x4 -2x3 + 3x2 - ax + 3a - 7 when divided by x

1.	The polynomial p{x) = x4 -2x3 + 3x2 - ax + 3a - 7 when divided by x+1 leaves the remainder 19. Find the values of a. Also, find the remainder when p(x) is divided by x+ 2.
Answer» p(x) = x⁴ – 2x³ + 3x² – ax + 3a – 7. Divisor = x + 1 x + 1 = 0 x = -1 So, substituting the value of x = – 1 in p(x), we get, p(-1) = (-1)⁴ – 2(-1)³ + 3(-1)² – a(-1) + 3a – 7. 19 = 1 + 2 + 3 + a + 3a – 7 19 = 6 – 7 + 4a 4a – 1 = 19 4a = 20 a = 5 Since, a = 5. We get the polynomial, p(x) = x⁴ – 2x³ + 3x² – (5)x + 3(5) – 7 p(x) = x⁴ – 2x³ + 3x² – 5x + 15 – 7 p(x) = x⁴ – 2x³ + 3x² – 5x + 8 As per the question, When the polynomial obtained is divided by (x + 2), We get, x + 2 = 0 x = – 2 So, substituting the value of x = – 2 in p(x), we get, p(-2) = (-2)⁴ – 2(-2)³ + 3(-2)² – 5(-2) + 8 ⇒ p(-2) = 16 + 16 + 12 + 10 + 8 ⇒ p(-2) = 62 Therefore, the remainder = 62.

The polynomial p{x) = x4 -2x3 + 3x2 - ax + 3a - 7 when divided by x+1 leaves the remainder 19. Find the values of a. Also, find the remainder when p(x) is divided by x+ 2.

Answer»

p(x) = x⁴ – 2x³ + 3x² – ax + 3a – 7.

Divisor = x + 1

x + 1 = 0

x = -1

So, substituting the value of x = – 1 in p(x), we get,

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

19 = 1 + 2 + 3 + a + 3a – 7

19 = 6 – 7 + 4a

4a – 1 = 19

4a = 20

a = 5

Since, a = 5.

We get the polynomial,

p(x) = x⁴ – 2x³ + 3x² – (5)x + 3(5) – 7

p(x) = x⁴ – 2x³ + 3x² – 5x + 15 – 7

p(x) = x⁴ – 2x³ + 3x² – 5x + 8

As per the question,

When the polynomial obtained is divided by (x + 2),

We get,

x + 2 = 0

x = – 2

So, substituting the value of x = – 2 in p(x), we get,

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

⇒ p(-2) = 16 + 16 + 12 + 10 + 8

⇒ p(-2) = 62

Therefore, the remainder = 62.