WEBVTT
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A polynomial π of π₯ is divided by π₯ minus π.
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Given that π₯ minus π is not a factor of π of π₯, what is the remainder equal to?
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Well, there is a theorem called the polynomial remainder theorem, or just the remainder theorem, that says that the remainder when dividing a polynomial π of π₯ by the linear polynomial π₯ minus π is π of π.
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So using the theorem, we see that the answer is just π of π.
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This theorem is very useful in practice because it allows us to find the remainder without going through the whole process of long division.
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We can prove this theorem by using what we know about the polynomial long division.
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When we use polynomial long division to divide a polynomial π of π₯ by π₯ minus π, essentially what weβre doing is rewriting it in the form π₯ minus π times π of π₯, the quotient, plus π of π₯, the remainder.
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In general, this remainder, π of π₯, is a polynomial in π₯.
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However, we know that the degree of the polynomial π of π₯ is always less than the degree of the polynomial that weβre dividing by.
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In this case, π₯ minus π.
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So we know that π of π₯ is just a constant polynomial, and weβll write it just as π.
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Okay.
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So we know that the remainder is a constant.
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I suppose thatβs slightly helpful.
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But we still need to know what that constant is.
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This equality holds for all values of π₯.
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So we can substitute whatever value of π₯ weβd like in there, and it will still be true.
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We can choose to substitute in π, and weβll get that π of π is equal to π minus π times π of π plus π.
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And of course, π minus π is just zero.
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So we get zero π of π plus π which is just π.
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So thatβs the proof of the theorem that weβve just used.
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And we can see that it works even if π₯ minus π is a factor of π of π₯, in which case π will just be zero.
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So when a polynomial π of π₯ is divided by π₯ minus π, given that π₯ minus π is not a factor of π of π₯ and even when it is in fact, the remainder is equal to π of π.