Lecture 22: Long division and zeros of polynomials

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Problem of the day

  • See Problem 10, due March 26
  • See Problem 11, due April 2

    Some Old POWs:

    The sum of the coefficients of (x-2)^2 = 1x^2-4x+4 is 1-4+4=1. What is the sum of the coefficients in the expanded form of (2x-3y)^5?

    If a Cardinal can pray a soul out of purgatory, by himself, in 1 hour, a Bishop in 3 hours, a Priest in 5 hours, and a Friar in 7 hours, how long would it take them to pray 4 souls out of purgatory, all praying together?

    Brief review of the lecture

    The Binomial Theorem can be used to expand powers of binomials (hence the name) without recourse to polynomial multiplication or Pascal's triangle.

    The rth entry in the nth row of Pascal's triangle is given by the formula

    nCr = n! / [(n-r)!r!]

    The number nCr is called a Binomial Coefficient. We looked a second time at several binomial expansions: (x+1)^5, (x+y)^5, (2x+1)^4, and (2-3)^4. We also looked at one more, which comes up in probability (1/3 + 2/3)^5. If you read this web page before class, you would profit from practicing these problems before the lecture.

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