Statement 1: The polynomial $x^{5} + 2 x^{4} + 3 x^{3} + x^{2} - 7 x + 8$ when divided by x + 1, gives 14 as remainder.

Statement 2: The value of the polynomial $x^{5} + 2 x^{4} + 3 x^{3} + x^{2} - 7 x + 8$ at x = -1 is 14.

(a) Both the statements are true and statement 2 is the correct explanation of statement 1..
(b) Both the statements are true and statement 2 is not the correct explanation of statement 1.
(c) Statement 1 is true but statement 2 is false.
(d) Statement 1 is false but statement 2 is true.
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Solution

Answer: Option (a)
Remainder theorem states that a polynomial P(x) of degree greater than or equal to one, when divided by x – a, gives P(a) as the remainder. P(a) should be a polynomial of degree greater than 1 and a can be any real number. So, when $x^{5} + 2 x^{4} + 3 x^{3} + x^{2} - 7 x + 8$ is divided by x +1, the remainder is P(-1) which is 14.
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The polynomial x^{5} + 2 x^{4} + 3 x^{3} + x^{2} - 7 x + 8 when divided by (x + 1) leaves a remainder of 14 .

The value of the polynomial x^{5} + 2 x^{4} + 3 x^{3} + x^{2} - 7 x + 8 at x = - 1 is 14.

The value of the polynomial $x^{5} + 2 x^{4} + 3 x^{3} + x^{2} - 7 x + 8$ at $x = - 1$ is 14.

Statement 1: The polynomial $P (x) = 4 x^{3} - 3 x^{2} + 5 x - 6$ when divided by $x - 1$ gives zero as the remainder.

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