Notes from Monday ...

Yesterday's lesson was on Binomial Theorem, which is a short cut for raising binomials to exponents. We simplify this process as much as we can. One way we simplify this is to use Pascal's Triangle. This "triangle" is an arrangement of numbers that continues forever. Each number is made by adding together the two numbers above it. Here is the triangle through the first 5 rows (the top number is considered to be row 0). The 6th row would be 1, 6, 15, 20, 15, 6, 1.


Example: Simplify (x + y)⁴

This problem would require us to multiply out (x + y)*(x + y)*(x + y)*(x + y). The students saw through a problem in class that this is a lengthy process with many places to have errors. The Binomial Theorem and Pascal's Triangle makes this much simpler.

1. Note the exponent and go to that row of Pascal's Triangle. Here, the exponent is 4 so we will use row 4: 1, 4, 6, 4, 1. These numbers become our coefficients (numbers in front of the variables). We write the numbers down followed by 2 blanks each: 1_ _ 4 _ _ 6 _ _ 4 _ _ 1_ _
2. To start filling in the blanks, use the first term inside the parenthesis (here, x). This term will start with the exponent of the problem (here, 4) and then count down as it fills in the blanks from left to right: 1 x⁴ _ 4 x³ _ 6 x² _ 4 x¹ _ 1 x⁰ _
3. To finish filling in the blanks, use the second term inside the parenthesis (here, y). This term starts with the exponent of 0 and counts up to the exponent of the problem as it fills in the blanks from left to right: 1 x⁴ y⁰ 4 x³ y¹ 6 x² y² 4 x¹ y³ 1 x⁰ y⁴ . Notice that each term has exponents that add together to 4 (the exponent in the problem).
4. Simplify, if possible. Remember, any non-zero value raised to the 0 power is 1, so we remove that from our answer. If there are any numbers raised to powers, this is the step to simplify those powers and multiply times the coefficients. Here: 1 x⁴ 4 x³ y¹ 6 x² y² 4 x¹ y³ 1 y⁴
5. Enter the signs throughout the polynomial answer. If the binomial has addition, then all signs in the answer are plus signs. If the binomial has subtraction, then all signs alternate plus and minus, with the first term being positive. Here: 1 x⁴ + 4 x³ y¹ + 6 x² y² + 4 x¹ y³ + 1 y⁴

Homework: Page 75, #2 - 16 even and #24

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