\[ f(x) = q(x) d(x) + r(x) \]

When \(d(x) \) is a polynomial with no repeated roots, Prasun has stated how we could determine \(r(x) \) using the Remainder-Factor Theorem.

How should we deal with the case when \(d(x)\) has repeated roots?

What is the remainder when \( x^{10} \) is divided by \( (x-1) ^2 \)?

What is the remainder when \( x^{10} \) is divided by \( (x-1) ^2 (x+1) \)?

## Comments

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TopNewestDifferentiate throughout and evaluate the expression at the value of the repeated root. – Sudeep Salgia · 2 years, 2 months ago

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Is there another simpler way?

Hint:How can we apply partial fractions cover up rule to repeated roots? – Calvin Lin Staff · 2 years, 2 months agoLog in to reply

\(f(x) = x^n\)

\(f(1) = 1\)

\(f(x) = (x - 1)Q(x) + 1\)

\(f'(x) = Q(x) + (x - 1)Q'(x)\)

\(f'(x) = nx^{n-1}\)

\(f'(1) = n\)

\(Q(1) = n\)

\(Q(x) = (x - 1)q(x) + n\)

\(f(x) = (x - 1)((x - 1)q(x) + n)) + 1\)

\(f(x) = (x - 1)^{2}q(x) + nx - n + 1\)

\(f(x) = (x - 1)^{2}(x + 1)q_1(x) + a(x - 1)^{2} + nx - n + 1\)

\(f(x) = x^{10}\)

\(f(-1) = 1\)

\(4a - 19 = 1\)

\(a = 5\)

– 汶良 林 · 2 years, 1 month agoLog in to reply

What does this remind you of?Hint:What is the remainder when \( f(x) \) is divided by \( (x-a) ^3 \)@Sudeep Salgia @Prasun Biswas The above is a useful way to remember how to find the remainder when divided by a linear power. It forms the linkage between remainders of a polynomial, and how the polynomial looks like at a point. – Calvin Lin Staff · 2 years, 1 month ago

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– Calvin Lin Staff · 2 years, 1 month ago

Close, but I'm thinking of something deeper. What would the answer to the hint be? That should be a 1 line answer with the correct "reminder".Log in to reply

Wait, IM SO CLOSE I MIGHT HAVE FIGURED IT OUT – Cs ಠ_ಠ Lee · 2 years, 1 month ago

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– Calvin Lin Staff · 2 years, 1 month ago

Please avoid typing in all capital letters, as that is considered rude on the internet.Log in to reply

\(x^n = ((x - a) + a)^n\)

– 汶良 林 · 2 years, 1 month agoLog in to reply

Find the "one-line" explanation for it. – Calvin Lin Staff · 2 years, 1 month ago

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What is a one-line explanation for the answer? – Calvin Lin Staff · 2 years, 1 month ago

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– 汶良 林 · 2 years, 1 month ago

\(x^n - (x - a)^n\)Log in to reply