Binomial Expansion For AS Level Maths
Binomial Expansion For AS Level Maths – Introduction
In this article we are going to look at 7 questions regarding the Binomial Expansion for AS Level Maths. You will meet this topic in years 12 and 13. You need to make sure that you are aware of the various techniques that exist for both year groups.
For year 12 you will deal with positive powers and also meet the NCR approach as well as learning about Pascal’s Triangle.
For year 13 you will deal with negative and fractional powers as well as use various techniques to expand partial fractions.
We will look at 10 key questions that cover Binomial Expansion for AS Level Maths for students in the year 12 group. Another article will cover the year 13 group at a future date.
When doing the NCR approach you need to remember the following formula:
(a+b)^n=a^n+\left(\begin{array}{l} n \\ 1 \end{array}\right) a^{n-1} b+\ldots+\left(\begin{array}{l} n \\ r \end{array}\right) a^{n-r} b^r+\ldots+b^n .Binomial Expansion For AS Level Maths - 3 Warm Up Questions
Since you only want to find the first three terms then you should consider using the NCR approach. You could do the question with Pacal, but you would need to produce a triangle with 5 rows which is not an issue, but it can be if the power of expansion is quite high. For questions like this you want to adopt the NCR approach as it is much faster.
\begin{aligned} (3+2 x)^5 & =3^5+\left(\begin{array}{l} 5 \\ 1 \end{array}\right) 3^4(2 x)^1+\left(\begin{array}{l} 5 \\ 2 \end{array}\right) 3^3(2 x)^2+\ldots \\ & =243+810 x+1080 x^2+\ldots \end{aligned}2. 
Once again use the NCR approach. Note that there is an unknown and this should not be off putting. SImply proceed as you would if there was a number.
\begin{aligned} (1+p x)^{12} & =1+\left(\begin{array}{c} 12 \\ 1 \end{array}\right)(p x)^1+\left(\begin{array}{c} 12 \\ 2 \end{array}\right)(p x)^2+\ldots \\ & =1+12 p x+66 p^2 x^2+\ldots \end{aligned}
Here you need to use the answer in part a) to help with this question. Effectively you are solving simultaneous equations.
3. 
Again, this question has a high power. Using Pascal whilst possible would take longer and is prone to errors in arithmetic. Using the NCR approach we have:
\begin{aligned} (1+p x)^9 & =1+\left(\begin{array}{l} 9 \\ 1 \end{array}\right)(p x)^1+\left(\begin{array}{l} 9 \\ 2 \end{array}\right)(p x)^2+\ldots \\ & =1+9 p x+36 p^2 x^2+\ldots \end{aligned}
Using the answer to part a) you can determine the value of p and from this the value of q can then be obtained.
\begin{gathered} 9 p=36 \Rightarrow \underline{\underline{p=4}} \\ q=36 p^2=36 \times 4^2=\underline{\underline{576}} . \end{gathered}Binomial Expansion For AS Level Maths - Further 2 Questions
4. 
Using the NCR approach we have:
\begin{aligned} (1+k x)^6 & =1+\left(\begin{array}{l} 6 \\ 1 \end{array}\right)(k x)^1+\left(\begin{array}{l} 6 \\ 2 \end{array}\right)(k x)^2+\left(\begin{array}{l} 6 \\ 3 \end{array}\right)(k x)^3+\ldots \\ & =1+6 k x+15 k^2 x^2+20 k^3 x^3+\ldots \end{aligned}
You are told that the coefficients are equal so they can be equated which then leads to a quadratic term, which can be factorised. Remember to read the question carefully as the value of K is not zero.
\begin{aligned} 6 k=15 k^2 & \Rightarrow 15 k^2-6 k=0 \\ & \Rightarrow 3 k(5 k-2)=0 \\ & \Rightarrow 5 k-2=0(\text { as } k \neq 0) \\ & \Rightarrow k=\frac{2}{5} . \end{aligned}5. 
Adopting the NCR approach we have:
\begin{aligned} \left(1+\frac{1}{2} x\right)^{10} & =1+\left(\begin{array}{c} 10 \\ 1 \end{array}\right)\left(\frac{1}{2} x\right)^1+\left(\begin{array}{c} 10 \\ 2 \end{array}\right)\left(\frac{1}{2} x\right)^2+\left(\begin{array}{c} 10 \\ 3 \end{array}\right)\left(\frac{1}{2} x\right)^3+\ldots \\ & =1+5 x+\frac{45}{4} x^2+15 x^3+\ldots \end{aligned}
To do this question you need to find the value of x that will give 1.005
\begin{aligned} & 1+0.5 x=1.005 \\ & x=0.01 \end{aligned}Using the expansion we can then obtain:
\begin{aligned} (1.005)^{10} & =\left(1+\frac{1}{2} \times 0.01\right)^{10} \\ & \approx 1+5(0.01)+\frac{45}{4}(0.01)^2+15(0.01)^3 \\ & =1.051137 \\ & =1.05114(5 \mathrm{dp}) \end{aligned}
Binomial Expansion For AS Level Maths – Final 2 Questions
6. 
For this question you can use either NCR or Pascal as the power is small. For this particular article the NCR method has been adopted as follows:
\begin{aligned} (2+3 x)^4 & =2^4+\left(\begin{array}{l} 4 \\ 1 \end{array}\right) 2^3(3 x)^1+\left(\begin{array}{l} 4 \\ 2 \end{array}\right) 2^2(3 x)^2+\left(\begin{array}{l} 4 \\ 3 \end{array}\right)(2)(3 x)^3+(3 x)^4 \\ & =16+96 x+216 x^2+216 x^3+81 x^4 . \end{aligned}
This is quite similar to the above but the only difference is the sign. When doing expansions you need to be very careful with signs especially when raising them to a power.
(2-3 x)^4=16-96 x+216 x^2-216 x^3+81 x^4 \text {. }7. 
As the power is fairly high, adopt the NCR method for this particular question as follows:
\begin{aligned} (2-9 x)^4 & =[2+(-9 x)]^4 \\ & =2^4+\left(\begin{array}{l} 4 \\ 1 \end{array}\right) 2^2(-9 x)^1+\left(\begin{array}{l} 4 \\ 2 \end{array}\right) 2^8(-9 x)^2+\ldots \\ & =16-288 x+1944 x^2+\ldots \end{aligned}
Now the question has become a little more complicated. You can expand the brackets using the terms that are most relevant and this technique is known as comparing coefficients. Now because A is independent of x then it must be 16. Below is a grid method technique that shows the values when the two brackets are multiplied together.
\begin{array}{c|ccc} \hline & 16 & -288 x & +1944 x^2 \\ \hline 1 & 16 & -288 x & +1944 x^2 \\ +k x & +16 k x & -288 k x^2 & \ldots \\ \hline \end{array}
To find the value of “k” you can simply compare the coefficients of “x” as shown below:
16 k-288=-232 \Rightarrow 16 k=56 \Rightarrow k=\frac{7}{2} .
To find B you need to consider the x2 terms. From the multiplication grid you can see this as follows and substituting in the value of “k”
B=1944-288 \times \frac{7}{2} \Rightarrow \underline{\underline{B=936}} .Binomial Expansion For AS Level Maths can and does involve some easy questions and some more challenging questions. The challenging questions do require more problem solving skills as well as more algebraic skills in order to solve them.
Try the above questions again on your own, making sure that you are understanding the solution that we have used. It is also important to know when to use Pascal compared to the NCR Binomial approach.