slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Chapter 8: Further Topics in Algebra PowerPoint Presentation
Download Presentation
Chapter 8: Further Topics in Algebra

Loading in 2 Seconds...

play fullscreen
1 / 17

Chapter 8: Further Topics in Algebra - PowerPoint PPT Presentation


  • 131 Views
  • Uploaded on

Chapter 8: Further Topics in Algebra. 8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 The Binomial Theorem 8.5 Mathematical Induction 8.6 Counting Theory 8.7 Probability. 8.4 The Binomial Theorem. The binomial expansions

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

Chapter 8: Further Topics in Algebra


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
    Presentation Transcript

    1. Chapter 8: Further Topics in Algebra 8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 The Binomial Theorem 8.5 Mathematical Induction 8.6 Counting Theory 8.7 Probability

    2. 8.4 The Binomial Theorem The binomial expansions reveal a pattern.

    3. 8.4 A Binomial Expansion Pattern • The expansion of (x + y)n begins with x n and ends with y n . • The variables in the terms after x n follow the pattern x n-1y , x n-2y2 , x n-3y3and so on to y n . With each term the exponent on x decreases by 1 and the exponent on y increases by 1. • In each term, the sum of the exponents on x and y is always n. • The coefficients of the expansion follow Pascal’s triangle.

    4. 8.4 A Binomial Expansion Pattern Pascal’s Triangle Row

    5. 8.4 Pascal’s Triangle • Each row of the triangle begins with a 1 and ends with a 1. • Each number in the triangle that is not a 1 is the sum of the two numbers directly above it (one to the right and one to the left.) • Numbering the rows of the triangle 0, 1, 2, … starting at the top, the numbers in row n are the coefficients of x n, x n-1y , x n-2y2 , x n-3y3, …y n in the expansion of (x + y)n.

    6. 8.4 n-Factorial n-Factorial For any positive integer n, and Example Evaluate (a) 5! (b) 7! Solution (a) (b)

    7. 8.4 Binomial Coefficients Binomial Coefficient For nonnegative integers n and r, with r<n,

    8. 8.4 Binomial Coefficients • The symbols and for the binomial coefficients are read “n choose r” • The values of are the values in the nth row of Pascal’s triangle. So is the first number in the third row and is the third.

    9. 8.4 Evaluating Binomial Coefficients Example Evaluate (a) (b) Solution (a) (b)

    10. 8.4 The Binomial Theorem Binomial Theorem For any positive integers n,

    11. 8.4 Applying the Binomial Theorem Example Write the binomial expansion of . Solution Use the binomial theorem

    12. 8.4 Applying the Binomial Theorem

    13. 8.4 Applying the Binomial Theorem Example Expand . Solution Use the binomial theorem with and n = 5,

    14. 8.4 Applying the Binomial Theorem Solution

    15. 8.4 rth Term of a Binomial Expansion rth Term of the Binomial Expansion The rth term of the binomial expansion of (x + y)n, where n>r – 1, is

    16. 8.4 Finding a Specific Term of a Binomial Expansion. Example Find the fourth term of . Solution Using n = 10, r = 4, x = a, y = 2b in the formula, we find the fourth term is