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Understanding the Binomial Theorem

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Questions and Answers

What is the Binomial Theorem used to expand?

(𝑎 + 𝑏)𝑛 when n is a negative integer

(𝑎 + 𝑏)𝑛 when n is a rational integer

(𝑎 + 𝑏)𝑛 when n is a prime number

(𝑎 + 𝑏)𝑛 when n is a positive integer (correct)

According to the binomial theorem, what is the expansion of (𝑎 + 𝑏)𝑛?

$rac{n}{2} 𝐶𝑟 𝑏 𝑟 𝑎𝑛−𝑟$

$𝑎^n + 𝑏^n$

$𝐶𝑟 𝑏 𝑟 𝑎𝑛−𝑟$ (correct)

$𝐶𝑟 𝑏 𝑟 𝑎^n−r$

What does the sum of the exponents of a and b in each term equal in the binomial expansion?

$2n$

$n-1$

$n$ (correct)

$n^2$

When should the infinite series be used to expand (𝑎 + 𝑏)𝑛 according to the definition?

When n is a negative integer Signup and view all the answers

How many terms does the binomial expansion have according to the binomial theorem?

$n+1$ Signup and view all the answers

What is the expansion of (2𝑥 − 3𝑦)4?

(2𝑥)4 + 4𝐶1(−3𝑦)1(2𝑥)3 + 4𝐶2(−3𝑦)2(2𝑥)2 + 4𝐶3(−3𝑦)3(2𝑥)1 + (−3𝑦)4 Signup and view all the answers

What is the general term in the expansion of (𝑎 + 𝑏)^n, where 𝒓 = 0, 1, 2, … , 𝒏?

$T_{r+1} = nC_r b^r a^{n-r}$ Signup and view all the answers

Find the fourth term in the expansion of (𝑥 − 1)^9.

$-84x^6$ Signup and view all the answers

If n = 4, what are the orders of the middle terms in the expansion of (3𝑥 + 7𝑦)^8?

4, 6 Signup and view all the answers

Is there a term that does not contain 𝒙 (a constant term or a term free of 𝒙) in the expansion of (3𝑥 − 𝟐)^8?

$90720$ Signup and view all the answers