At the end of this chapter, you will be able to:
- know what is Combination and how to use it in Binomial expansions,
- know how to find a specific term Tr+1 of a Binomial expansion,
- know how to apply Binomial Theorem to expand (a+b)n,
- know all you need to know about Binomial Theorem.
Types of questions
- Given term independent of x, find unknown using general term Tr+1
- Expand up to the required term and hence evaluate a number
- Find the first few terms and hence obtain coefficient of a specific term
- Given coefficient of a term of the product of two binomials, find unknown
- Given first few terms of the product of two binomials, find unknowns
- Given first few terms of a binomial to the power of n, find unknowns
- Given first two non-zero terms of the product of two binomials, find unknowns
Term independent of x refers to the constant term. The skill involved here requires the use of the general term Tr+1 and may be applied to find any other terms (x, x2… etc).
The number you are required to evaluate is normally similar to the binomial you have just expanded. The key here is to determine what value x should be by setting everything in the bracket equals to each other. After which you substitute that value of x into what you have just expanded and you will find your answer.
This question is sort of guided in the sense that they direct you to the number of terms you need to expand in order to obtain the coefficient you need. The main idea here is to compare coefficients after the binomial expansion.
If you have realised perhaps, this question is actually very similar to the previous question. The only difference is they switched the unknown, meaning you have to find a different thing here. The skill required however is the same. Compare coefficient after the binomial expansion.
This is also pretty much the same as the previous question, just that instead of only comparing the coefficients of one term, you will have to compare more terms. This will also mean the likelihood of solving simultaneous equations, but it is actually very simple.
The difference between this and the previous question is that the unknowns have been switched. Now the power of the binomial is an unknown, meaning you will have to utilise the concept of Combination since you cannot use the calculator in this situation. Otherwise it is actually similar to the previous questions, expand and compare coefficients, solve simultaneous equations if needed.
This is just another variation of the same concept throughout this topic. By telling you that the first two non-zero terms are the constant term and the x2 term, they are just going one big round to tell you that the coefficient of the x term is 0 that’s all. Nothing novel. Expand, compare coefficients and solve simultaneous equations if needed.
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