Pre-Cal 40S May 12, 2009
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The document discusses binomial expansions and combinatorics. It explains the patterns in binomial expansions algebraically and combinatorially. Specifically, it outlines the relationships between the exponents of terms and their coefficients as given by Pascal's triangle and binomial coefficients. The document provides examples of finding individual terms in binomial expansions.
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![The Binomial Theorem ...
Algebraically
Combinatorically
Notice the patterns ...
(1) The coefficient of the term is:
(2) The exponent on a is given by: [n - (i - 1)]
(3) The exponent on b is given by: i - 1
(4) This relation holds for each term in the expansion:
[exponent on a] + [exponent on b] = n
(5) The number of terms in any binomial expansion is: n + 1](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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2) The degree of a monomial is the sum of the exponents of its variables. The degree of a polynomial is the highest degree among its monomial terms.
3) Polynomials are classified by degree, with linear having degree 1, quadratic degree 2, cubic degree 3, and so on. Basic arithmetic with polynomials involves combining like terms.
11.4 The Binomial Theorem
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
The Binomial Theorem.pptx
The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It explains that the terms follow certain patterns, with the exponents of the variables alternating and summing to n. The coefficients of the terms form Pascal's triangle. The binomial theorem states that the expansion of (a + b)^n is the sum of terms with coefficients given by the binomial coefficients. Examples are provided to demonstrate expanding binomials and finding particular terms using the binomial theorem formula.
Pc 2.1 notes
This document defines quadratic equations as polynomial functions of degree 2, where the highest exponent is 2. It explains that a quadratic equation can be written in standard form as f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a is not equal to 0. The graph of a quadratic equation is a parabola. Key features of a parabola include its axis of symmetry, which is the line that bisects the parabola, and its vertex, which is the highest or lowest point of the parabola. The document provides examples of finding the vertex and axis of symmetry of parabolas given their equations and of writing the equation of a parabola
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- 1. Poker Combinatorics or How likely is THAT?!? Red Hot Poker by flickr user kimberlyfaye
- 2. The Binomial Theorem ... Algebraically Combinatorically Notice the patterns ... (1) The coefficient of the term is: (2) The exponent on a is given by: [n - (i - 1)] (3) The exponent on b is given by: i - 1 (4) This relation holds for each term in the expansion: [exponent on a] + [exponent on b] = n (5) The number of terms in any binomial expansion is: n + 1
- 3. Evaluate each term ... n! nC r = (n-r)!r!
- 4. Pascal's Triangle Can you find the Hockey Stick pattern?
- 5. Any individual term, let's say the ith term, in a binomial expansion can be represented like this: Example: Find the 4th term in the expansion of
- 6. Determine the indicated term in each expansion. (a) the 8th term in the expansion of
- 7. Example: Find the term that contains x7 in the expansion of
- 8. Find the term that contains in the expansion of:
- 9. Find the term that contains in the expansion of: