The polynomial p(x) = x4 - 2x3 + 3x2 - 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) = x4 - 2x3 + 3x2 - 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) = x4 - 2x3 + 3x2 - 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) = x4 - 2x3 + 3x2 - 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) = x4 - 2x3 + 3x2 - ax + 3a - 7 is divided by x + 2, are 5 and 62 respectively.
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