# The polynomial p(x) = x^{4} - 2x^{3} + 3x^{2} - ax + 3a - 7 when divided by x + 1 leaves the remainder 19. Find the value of a. Also find the remainder when p(x) is divided by x+2.

**Solution: **

**The remainder theorem is stated as follows: **

**When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k).**

**Let us use the remainder theorem to find the remainder when the polynomial p(x) = x ^{4} - 2x^{3} + 3x^{2} - ax + 3a - 7 is divided by x + 1.**

Let x + 1 = 0

⇒ x = -1

By using Remainder theorem,

p (-1) = 19

(-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 = 5

Let us find the value of p (x) = x^{4} - 2x^{3} + 3x^{2} - ax + 3a - 7 when divided by x + 2.

Let (x + 2) = 0

⇒ x = -2

p (-2) = (-2)^{4} - 2 (-2)^{3} + 3(-2)^{2} - (5) (-2) + 3(5) - 7

⇒ p(-2) = 16 + 16 + 12 + 10 + 15 - 7

⇒ p(-2) = 62

## The polynomial p (x) = x^{4} - 2x^{3} + 3x^{2} - ax + 3a - 7 when divided by x + 1 leaves the remainder 19. Find the value of a. Also find the remainder when p(x) is divided by x+2.

**Summary:**

The values of ‘a’ and p(x), when the polynomial p (x) = x^{4} - 2x^{3} + 3x^{2} - ax + 3a - 7 is divided by x + 2, are 5 and 62 respectively.

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