Question

Statement 1:
$\text{The polynomial}{x}^5+2x^4+3x^3+x^2–7x+8\text{when divided by}(x+1)\text{leaves a}\text{remainder of 14}.$

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

• 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. Both statements are false.

Answer 1

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

Given polynomial:
$p\left(x\right)=x^5+2x^4+3x^3+x^2–7x+8p\left(x\right)\text{is divided by}(x+1)$

$\text{Then,}x+1=0x=-1$

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

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

$p(-1)=14$
So, statement 2 is correct.

$\text{By remainder theorem, if a}\text{polynomial}f\left(x\right),\text{is divided by}(x-a),\text{it's remainder is given by}f\left(a\right).$

$p(-1)\text{is the remainder when}p\left(x\right)\text{is divided by}(x+1).$

So, statement 1 is correct and is explained by statement 2.