6.The polynomial p(x) = x4-2x3 + 3x2-ax + 30-7 when divided by x + 1,

1.	6.The polynomial p(x) = x4-2x3 + 3x2-ax + 30-7 when divided by x + 1, leaves theremainder 19. Find the value of a. Also, find the remainder when p(x) is divided byx +2.
Answer» Given Polynomial⇒ P(x) = x⁴ - 2x³ + 3x² - ax + 3a - 7. Divisor = x + 1∴ x + 1 = 0∴ x = -1 Thus, P(-1) = (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + 3a - 7.19 = 1 + 2 + 3 + a + 3a - 719 = 6 - 7 + 4a4a - 1 = 194a = 20⇒ a = 4 ∴ Value of a is 4. Now, the Polynomial will be ---→ P(x) = x⁴ - 2x³ + 3x² - (4)x + 3(4) - 7P(x) = x⁴ - 2x³ + 3x² - 4x + 12 - 7P(x) = x⁴ - 2x³ + 3x² - 4x + 5 Now, When this polynomial is divided by (x + 2), then, x + 2= 0x = - 2 ∴ P(-2) = (-2)⁴ - 2(-2)³ + 3(-2)² - 4(-2) + 5⇒ P(-2) = 16 + 16 + 12 + 8 + 5⇒ P(-2) = 57 Thus, Remainder will be 57.

6.The polynomial p(x) = x4-2x3 + 3x2-ax + 30-7 when divided by x + 1, leaves theremainder 19. Find the value of a. Also, find the remainder when p(x) is divided byx +2.

Answer»

Given Polynomial⇒

P(x) = x⁴ - 2x³ + 3x² - ax + 3a - 7.

Divisor = x + 1∴ x + 1 = 0∴ x = -1

Thus, P(-1) = (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + 3a - 7.19 = 1 + 2 + 3 + a + 3a - 719 = 6 - 7 + 4a4a - 1 = 194a = 20⇒ a = 4

∴ Value of a is 4.

Now, the Polynomial will be ---→

P(x) = x⁴ - 2x³ + 3x² - (4)x + 3(4) - 7P(x) = x⁴ - 2x³ + 3x² - 4x + 12 - 7P(x) = x⁴ - 2x³ + 3x² - 4x + 5

Now, When this polynomial is divided by (x + 2), then,

x + 2= 0x = - 2

∴ P(-2) = (-2)⁴ - 2(-2)³ + 3(-2)² - 4(-2) + 5⇒ P(-2) = 16 + 16 + 12 + 8 + 5⇒ P(-2) = 57

Thus, Remainder will be 57.