Question

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

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

• A. Statement 1 is false but statement 2 is true.
• B. Both the statements are true and statement 2 is the correct explanation of statement 1.
• C. Both the statements are true and statement 2 is not the correct explanation of statement 1.
• D. Statement 1 is true but statement 2 is false.
  

Answer 1

The correct option is B Both the statements are true and statement 2 is the correct explanation of statement 1.

In both statements $p\left(x\right)=x^5+2x^4+3x^3+x^2–7x+8$.

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.

So, when $p\left(x\right)=x^5+2x^4+3x^3+x^2–7x+8$ is divided by $x+1$, the remainder is $p(-1)$

$p(-1)={-1}^5+2{-1}^4+3{-1}^3+{-1}^2+7+8$

$p(-1)=-1+2-3+1+7+8$

$p(-1)=14$

we also know that, the value of $p\left(x\right)$ at $x=-1$ is $p(-1)$ which is 14.

Hence, Both the statements are true and statement 2 is the correct explanation of statement 1.