Solve for remainder

Algebra Level 3

Find the remainder when $x^{155}$ is divided by $x^2 + 3x +2.$

None of the other choices.

2^{155}

3^{154}

3^{155}

2 solutions

Chew-Seong Cheong
Aug 28, 2016

We note that $x^2 + 3x +2 = (x+1)(x+2)$ . Let $f(x) = x^{155}$ and by remainder factor theorem ,

Remainder of $f(x)$ divided by $x+1$ is $= f(-1) = (-1)^{155} = -1$ .
Similarly, remainder of $f(x)$ divided by $x+1$ is $= f(-2) = (-2)^{155} = -2^{155}$ .
Therefore, remainder of $f(x)$ divided by $(x+1)(x+2)$ is $= (-1)(x+2)+(2^{155})(x+1)$ $= (2^{155}-1)x - 2 + 2^{155}$ .

Therefore, the answer is $\boxed{\text{None of the other choices.}}$

From where did the 3rd point comes ? can u prove it.

Prince Raj - 4 years, 9 months ago

Sorry, it should be $R(x) = (-1)(x+2) + (\color{#D61F06}{+}2^{155})(x+1)$

It is a combination of statements 1 and 2. $R(x) = (-1)(x+2) + (2^{155})(x+1)$ . When $x=-1$ , $R(-1) = (-1)(-1+2) + (-2^{155})(0) = -1$ . When $x=-2$ , $R(-2) = (-1)(0) + (2^{155})(-1) = -2^{155}$ . The other $x$ also follows the linear $R(x)$ .

Chew-Seong Cheong - 4 years, 9 months ago

Shyam Upadhyay
Aug 28, 2016

Remainder will be linear Remainder=(2^155 -1)x +2^155 -2

Solve for remainder

2 solutions

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