Pascal's Triangle to expand the binomial

Extra Practice In Exercises 1-3, find the sum or difference. (-4 - 6x + 8) + (-7x + 8) Collect like terms. -4 - 6x + 8 -7x + 8 -5x² + x + 26 -12 + 48 - (-x² + 24x - 63) 6x² - 7x + 48 +x² - 24x + 63 7x² - 36x + 111 (-11x⁴ - x³ - 3x² + 10x - 2) - (-11x⁴ + 5x² - 7x + 13) -11x⁴ - x³ - 3x² + 10x - 2 +11x⁴ - 5x² + 7x - 13 -8x² + 17x - 15 In Exercises 4-9, find the product. 2x²(2x³ - x² + 3x - 5) 2x²[4x³ - 2x² + 6x - 10] = 4x⁵ - 2x⁴ + 6x³ - 10x² (x⁴ - 10x² + 25)(3x² - 6x - 1) Distribute x⁴ 3x² - 6x - 1 -10x² -30x² + 60x + 10 25 -75x² + 150x + 25 3x⁶ - 6x⁵ - 31x⁴ + 60x³ + 75x² + 150x - 25 Use box method only when multiplying it keeps things neat and organized In Exercises 4-9, find the product. (x + 1)(x - 2)(x + 6) box method multiplication = x³ + 5x² - 8x - 12 (2x - 3)(6 - x)(4 - 5x) box method multiplication = -10x³ + 75x² - 150x + 72 (3y - 8)(3y + 8) box method multiplication = 9y² - 64 (2v - 1)³ box method multiplication for (2v 1) * (2v 1) and then another box method for the result multiplied by (2v 1) = 8v³ - 12v² + 6v - 1 nice ... but there is a short cut when a binomial multiplies itself like this a number of times. Use the Pascal's Triangle In Pascal's Triangle, the first and last numbers in each row are 1. Every number other than 1 is the sum of the closest two numbers in the row directly above it. The numbers in Pascal's Triangle are the same numbers that are the coefficients of binomial expansions, as shown in the first six rows.   n ((a+b)^{n}) Pascal's Triangle Binomial Expansion   0th row 0 ((a+b)^{0}=)** 1 1 1st row 1 ((a+b)^{1}=)** 1 1 a + b 2nd row 2 ((a+b)^{2}=)** 1 2 1 a² + 2ab + b² 3rd row 3 ((a+b)^{3}=)** 1 3 3 1 a³ + 3a²b + 3ab² + b³ 4th row 4 ((a+b)^{4}=)** 1 4 6 4 1 a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ 5th row 5 ((a+b)^{5}=)** 1 5 10 10 5 1 a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵ 6 15 20 15 6 1 you can expand the triangle down as far as you want. If your binomial is being expanded (raised to a power), use the short cut above to multiply. 9. (2v - 1)³ - Use row 3 = 1a³ + 3a²b + 3ab² + 1b³ = 1(2v)³ + 3(2v)²(-1) + 3(2v)(-1)² + 1(-1)³ = 8v³ - 12v² + 6v - 1 Same answer as earlier using boxes. In Exercises 10 and 11, use Pascal's Triangle to expand the binomial. 10. (4t - 2)⁴ - Use row 4 = 1(4t)⁴ + 4(4t)³(-2) + 6(4t)²(-2)² + 4(4t)(-2)³ + 1(-2)⁴ = 1(256t⁴) + 4(64t³)(-2) + 6(16t²)(4) + 4(4t)(-8) + 1(16) = 256t⁴ - 512t³ + 384t² - 128t + 16 Seriously! easier than boxing it out !!! seriously 11. (g + 6)⁵ - Use row 5 = 1(g)⁵ + 5(g)⁴(6) + 10(g)³(6)² + 10(g)²(6)³ + 5(g)(6)⁴ + 1(6)⁵ = 1(g⁵) + 5(g⁴)(6) + 10(g³)(36) + 10(g²)(216) + 5(g)(1296) + 1(7776) = g⁵ + 30g⁴ + 360g³ + 2160g² + 6480g + 7776