By: Michelle Green. Construction of Pascal’s Triangle 1 0+1 1+1 4 1+2 0+1 41 6 1 10 55 1 1 ROW 0 ROW 1 ROW 2 ROW 3 ROW 4 ROW 5 20151 1 ROW 6 ALWAYS.

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Presentation transcript:

By: Michelle Green

Construction of Pascal’s Triangle ROW 0 ROW 1 ROW 2 ROW 3 ROW 4 ROW ROW 6 ALWAYS start with a 1 To fill in a number, add the numbers above it to the left and right

Patterns Within Pascal’s Triangle Horizontal Sums Prime Numbers Hockey Stick Exponents of 11

Horizontal Sums ROW 0 = 1 ROW 1 = 1+1=2 ROW 2 = 1+2+1=4 ROW 3 = =8 ROW 4 = = ROW 5= = 32 ROW 6= = 64 Can you see the pattern of the horizontal sums? BACK

Prime Numbers For any row that begins (not counting the one) with a prime number, all other numbers are divisible by the prime number. Examples: – Row 5: both 5 and 10 are divisible by 5 – Row 11: , 55, 165, 330, 462 are all divisible by 11 BACK

Hockey Stick Pattern BACK

Exponents of 11 BACK ⁰=1 11 ¹ =11 11 ² = ⁵ =??

Using Pascal’s Triangle Pascal’s Triangle can be used to expand binomials – (a+b)ⁿ For example we can use Pascal’s Triangle to expand: (x+3) ² –Since we know how to FOIL, we know the answer is x*x+3x+3x+9=x²+6x+9 – All we need to do is take a row from Pascal’s Triangle Always use the row of the exponent (here we need row 2 of Pascal’s Triangle)

Expanding (x+3) ² Take row 2 of Pascal’s Triangle – 1,2,1 – These will be our coefficients Starting with 2, decrease the exponent for x. Also, start with 0 and increase the exponent for 3. 1 (x) ²(3)º + 2(x)¹(3)¹ + 1(x)º(3)² x²+6x+9 Using the triangle may seem harder, but for higher exponent, it is easier than multiplying everything out.

Expand (2x+6)⁵ This binomial will be too time consuming to multiply out, so we will use Pascal’s Triangle. Take row 5 of Pascal’s Triangle – (2x)⁵(6)⁰ + 5(2x)⁴(6)¹ + 10(2x)³(6)² + 10(2x)²(6) ³+ 5(2x)¹(6)⁴ + 1(2x)⁰(6)⁵ 1(32x⁵)1 + 5(16x⁴)6 + 10(8x³) (4x²)(216) + 5(2x)(1296) + 1(1)(7776) 32x⁵ + 480x⁴ x³ x² x

Practice Using your Pascal’s Triangle, expand the following binomials: – (x+4)² – (5x+2)³ – (3x+1)³ – (2x+9)⁴

Resources Pascal’s Triangle and Its Patterns 16 October 2011.