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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 :)

Note by Timothy Wan
8 months, 3 weeks ago

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What is \(r\) in \(x^2+4r-5\)? Aditya Agarwal · 8 months, 3 weeks ago

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@Aditya Agarwal It's an error. Just corrected the note. Sorry! Timothy Wan · 8 months, 3 weeks ago

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Have you read the remainder factor theorem? If so, how do you think this could apply? Calvin Lin Staff · 8 months, 3 weeks ago

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@Calvin Lin 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 · 8 months, 3 weeks ago

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@Aditya Agarwal 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 · 8 months, 3 weeks ago

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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 · 8 months, 3 weeks ago

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@Timothy Wan You got confused between \(f(x)\) and \(r(x)\). Anupam Nayak · 8 months, 2 weeks ago

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