If the remainder when $x^{3} + 2 x^{2} + 3 x + 1$ is divided by $x^{2} + x + 1$ is $a x + b$ , then what is the value of $a + b$ ?

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Solution

The correct option is C 1
$x^{2} + x + 1 x + 1) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{3} + 2 x^{2} + 3 x + 1 x^{3} + x^{2} + x - - - ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{2} + 2 x + 1 x^{2} + x + 1 ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x$
Remainder $= x$
Comparing $x$ and $a x + b$ , we get $a = 1, b = 0$
Hence $a + b = 1$

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When $x^{4} + x^{3} - 2 x^{2} + x + 1$ is divided by x-1, the remainder is 2 and the quotient is q(x). Find q(x).

Divide $x^{5} - 4 x^{3} + x^{2} + 3 x + 6$ by $x^{3} - 3 x + 1$ ,and then find the remainder and quotient.

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= x - 1

= 2

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