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Remainder Theorem - Follow on

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

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 \( (x-1) ^2 (x+1) \)?

Note by Calvin Lin
2 years, 2 months ago

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

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@Sudeep Salgia That's one possible approach. However, as \(d(x) \) gets complicated, you could run into various issues. E.g. how would you evaluate the 2 given questions?

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 ago

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

\(f(1) = 1\)

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

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

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

\(f'(1) = n\)

\(Q(1) = n\)

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

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

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

The remainder when \( x^{10}\) is divided by \((x - 1)^{2}\) is \(10x - 9.\)

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

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

\(f(-1) = 1\)

\(4a - 19 = 1\)

\(a = 5\)

The remainder when \( x^{10}\) is divided by \((x - 1)^{2}(x + 1)\) is \(5(x - 1)^{2} + 10x - 9 = 5x^2 - 4.\)

汶良 林 · 2 years, 1 month ago

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@汶良 林 Yes, that is the right approach. More generally,

The remainder when \( f(x) \) is divided by \( (x-a) ^2 \) is \( f'(a) (x-a) + f(a) \).

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

Tangent line to \(f(x) = x^n\) at \((a, f(a)\)?
汶良 林 · 2 years, 1 month ago

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@汶良 林 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". Calvin Lin Staff · 2 years, 1 month ago

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

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@Cs ಠ_ಠ Lee Please avoid typing in all capital letters, as that is considered rude on the internet. Calvin Lin Staff · 2 years, 1 month ago

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

The remainder when \( x^{10}\) is divided by \((x - a)^3\) is \(_nC_2(x - a)^2 a^{n - 2} + _nC_1(x - a)^1 a^{n - 1} + a^n .\)

汶良 林 · 2 years, 1 month ago

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@汶良 林 What is the general solution, and why?

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

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@Calvin Lin \((( x - a) + a)^n = \sum_{i=0}^n nC_i (x - a)^{n - i} a^i\)

The remainder when \( x^n\) is divided by \((x - a)^k\) is \( \sum_{i=n-k+1}^n nC_i (x - a)^{n - i} a^i.\)

汶良 林 · 2 years, 1 month ago

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@汶良 林 What is the remainder when \( f(x) \) is divided by \( (x-a)^n \)?

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

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@Calvin Lin \(x^n - (x - a)^n\) 汶良 林 · 2 years, 1 month ago

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