The Binomial Theorem Assignment

Evaluate 0.985 using the Binomial Theorem. Write 0.98 as a binomial: (1 + ____) Enter the value for each variable. a = _ b = ____ n = _ Evaluate the first four terms in the binomial expansion. 1st term = 2nd term = 3rd term = 4th term =

-0.02 a = 1 b = -0.02 n = 5 1st term = 1 2nd term = -0.1 3rd term = 0.004 4th term = -0.00008

Evaluate 1.36 using the Binomial Theorem. Write 1.3 as a binomial: (1 + __) Complete the expansion of (a + b)6 with a = 1 and b = 0.3. (1 + 0.3)6 = (1)6 + 6(1)5(0.3) + 15(1)4(0.3)2+20(1)3(0.3)3+15(1)2(0.3)4 + 6(1)(0.3)5 + (0.3)6 After evaluating the powers, the expression reduces to =

.3 (1 + 0.3)6 = (1)6 + 6(1)5(0.3) + 15(1)4(0.3)2+20(1)3(0.3)3+15(1)2(0.3)4 + 6(1)(0.3)5 + (0.3)6 a: -----------0.000729

Explain how you can use the terms from the binomial expansion to approximate 0.985. 1st term = 1 2nd term = −0.1 3rd term = 0.004 4th term ≈ 0

Rashad's Response: There are 5 + 1 = 6 terms in the binomial expansion of (1−0.02)5, and since the 4th term is approximately 0, the 5th and 6th terms are also approximately 0. So, approximate the value of 0.985 by adding the first three terms: 1 + (-0.1) + 0.004 = 0.904.

Explain how you can use the terms from the binomial expansion to approximate 4.025. 1st term: 1024 2nd term: 25.6 3rd term: 0.256 4th term: 0.001

The 4th term is almost 0, so the 5th and 6th terms are also close to 0. Add the terms shown to approximate 4.025: 1024 + 25.6 + 0.256 + 0.001 = 1049.857.

Which expression represents the sixth term in the binomial expansion of (2a - 3b)10? 10C5(2a)5(-3b)5 10C5(-2a)5(3b)5 10C6(2a)4(-3b)6 10C6(-2a)4(3b)6 10C6(2a)6(-3b)4 10C6(-2a)6(3b)4

a: 10C5(2a)5(-3b)5

Which row of Pascal's triangle would you use to expand (2x + 10y)15? row 10 row 12 row 15 row 25 How many terms are in this expansion? __ terms What is the first term in this expansion? 2x15 215x15 10y15 1015y15

c: row 15 16 b: 2^15x^15

Use the binomial theorem to expand (a + b)6. 6C0a6 + 6C1a5b + 6C2a4b2 + 6C3a3b3 + 6C4a2b4 + 6C5ab5 6C0a6b + 6C1a5b2 + 6C2a4b3 + 6C3a3b4 + 6C4a2b5 + 6C5ab6 + 6C6b7 6C0a6 + 6C1a5b + 6C2a4b2 + 6C3a3b3 + 6C4a2b4 + 6C5ab5 + 6C6b6 Select the set of binomial coefficients for the expansion above.

c:6C0a6 + 6C1a5b + 6C2a4b2 + 6C3a3b3 + 6C4a2b4 + 6C5ab5 + 6C6b6 b: 1,6,15,20,15,6,1

What is the coefficient of the x6y3 term in the expansion of (x + 2y)9? 84 168 336 672

d: 672

What is the fifth term in the expansion of (3x - 3y)7? 76545 x3y4

76545x^3y^4

Explain why it would not make sense to approximate 256 using the binomial theorem for the binomial (20 + 5)6.

Because 5 is not between -1 and 1, when it is raised to a positive integer exponent it will not approach 0. Therefore, all seven terms would need to be calculated and added to find (20 + 5)6.

Without actually writing the formula, explain how to expand (x + 3)7 using the binomial theorem.

To write the coefficients of the 8 terms, either start with a combination of 7 things taken 0 at a time and continue to 7 things taken 7 at a time or use the 7th row of Pascal's triangle. For the first term, write x to the 7th power and 3 to the 0 power. Then decrease the power on x and increase the power on y until you reach x to the 0 and y to the 7. Simplify by evaluating the coefficients and powers of 3.

Which of the following is the expansion of (3c + d2)6? 729c6 + 1,458c5d2 + 1,215c4d4 + 540c3d6 + 135c2d8 + 18cd10 + d12 729c6 + 1,458c5d + 1,215c4d2 + 540c3d3 + 135c2d4 + 18cd5 + d6 729c6 + 1,215c5d2 + 810c4d4 + 270c3d6 + 90c2d8 + 15cd10 + d12 729c6 + 243c5d2 + 81c4d4 + 27c3d6 + 9c2d8 + 3cd10 + d12 c6 + 6c5d2 + 15c4d4 + 20c3d6 + 15c2d8 + 6cd10 + d12

a: 729c6 + 1,458c5d2 + 1,215c4d4 + 540c3d6 + 135c2d8 + 18cd10 + d12

Which of the following is the expansion of (2m - n)7? 128m7 + 448m6n + 672m5n2 + 560m4n3 + 280m3n4 + 84m2n5 + 14mn6 + n7 128m7 + 14m6n + 42m5n2 + 70m4n3 + 70m3n4 + 42m2n5 + 14mn6 + n7 m7 + 7m6n + 21m5n2 + 35m4n3 + 35m3n4 + 21m2n5 + 7mn6 + n7 128m7 - 384m6n + 480m5n2 - 320m4n3 + 160m3n4 - 60m2n5 + 12mn6 - n7 128m7 - 448m6n + 672m5n2 - 560m4n3 + 280m3n4 - 84m2n5 + 14mn6 - n7

e: 128m7 - 448m6n + 672m5n2 - 560m4n3 + 280m3n4 - 84m2n5 + 14mn6 - n7