A2t-pascalsbinomial

Pascal’s Triangle – Final Project Option Name ________________________ A LGEBRA II WITH T RIGONOMETRY Date ____________ Pd _________ In this project, you will investigate Pascal’s Triangle and its relationship to binomial coefficients. These topics bridge the gap between algebra, probability, geometry, and number theory. I.

Pascal’s Triangle (30pts) Create a large, poster-size Pascal’s Triangle. Be creative. Make a smaller, page-sized one first, to ensure correctness. Start with a 1 at the top center of a page. Around that 1, envision a sea of zeros. Each number in the next row will be the sum of the two numbers above it. If there’s only one number above, then it’s like adding zero, and the number below will be the same. The zeroth, first, and second rows are below. Check your third row with Mr. Yates to make sure that you are on the right track! 1 1

II.

1

2

1

1

Binomial Coefficients (60pts) 1. Write (a + b)2 as a polynomial in standard form. Show work (FOIL/distributive property). 2. Write (a + b)3 as a polynomial in standard form. Show work. 3. Write (a + b)4 as a polynomial in standard form. Show work (ask Mr. Yates for a hint if this seems too hard). 4. Compare the number of terms you get with the power used. What do you notice? 5. What patterns do you see in the exponents? 6. What patterns do you see in the coefficients? 7. Write (a + b)5 as a polynomial in standard form without multiplying / showing work. 8. Write (a + b)10 as a polynomial in standard form. 9. Read the example on p.292 about how expanding (x – 2)3. Then use this technique to expand (i.e. write in standard form) the polynomial (x – 2)4. 10. Expand (3x – y)4. 11. Expand (q + 5)7. 12. Expand (2m + 3n)6.

III.

More on the Triangle (10pts) Find and show another interesting fact about Pascal’s Triangle. Possibilities could include patterns within the triangle, the connection between Pascal’s Triangle and fractals like Sierpinski’s Triangle, or probability/combinatorics. You should include at least one paragraph (five sentences) written explanation, plus a visual demonstration of the interesting fact.