SMO 2015 Q10 Round 1

When a polynomial $f(x)$ is divided by $(x - 1)$ and $(x + 5)$ , the remainders are -6 and 6 respectively. Let $r(x)$ be the remainder when $f(x)$ is divided by $x^2 +4x - 5$ . Find the value of $r(-2)$ .

(A) 0 (B) 1 (c) 2 (D) 3 (E) 5

How does one do this sort of question?

I'm not experienced with polynomials, so it may seem a simple question to you but not to me. But please help! I'm a learner too :)

#Algebra

Easy Math Editor

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Comments

What is $r$ in $x^2+4r-5$ ?

Aditya Agarwal - 5 years, 5 months ago

It's an error. Just corrected the note. Sorry!

Timothy Wan - 5 years, 5 months ago

Have you read the remainder factor theorem? If so, how do you think this could apply?

Calvin Lin Staff - 5 years, 5 months ago

yeah you are right, but here, we would apply more of Euclids Division Algorithm for polynomials. Won't we? (Which essentially constitutes the proof of remainder theorem)

Aditya Agarwal - 5 years, 5 months ago

There are many equivalent ways of expressing the ideas involved in this question. I was pointing out one possible approach, which I think should be thought of given the phrasing of the question.

Calvin Lin Staff - 5 years, 5 months ago

Since the divisor is of degree 2, $r(x)$ is linear and in the form $Ax+B$ .

$f(1)=A(1)+B=A+B=-6$ .

Also $f(-5)=A(-5)+B=-5A+B=6$ .

Solving the simultaneous equations, $A=-2,B=-4$

$r(x)=-2x-4$

$\therefore r(-2)=-2(-2)-4=0$

Is this a correct solution?

Timothy Wan - 5 years, 5 months ago

You got confused between $f(x)$ and $r(x)$ .

Anupam Nayak - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$