3. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
$(2 - 3x)^6$ giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 5 - 2014 - Paper 1
Question 5
3. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
$(2 - 3x)^6$ giving each term in its simplest form.
(b) Hence, or otherwise,... show full transcript
Worked Solution & Example Answer:3. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
$(2 - 3x)^6$ giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 5 - 2014 - Paper 1
Step 1
Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $(2 - 3x)^6$
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Answer
To find the first three terms, we utilize the Binomial Theorem which states:
(a+b)n=∑k=0n(kn)an−kbk
In this case, let a=2, b=−3x, and n=6.
First Term (when k=0): T0=(06)(2)6(−3x)0=1⋅64⋅1=64
Second Term (when k=1): T1=(16)(2)5(−3x)1=6⋅32⋅(−3x)=−576x
Third Term (when k=2): T2=(26)(2)4(−3x)2=15⋅16⋅9x2=2160x2
Thus, the first three terms are:
64−576x+2160x2
Step 2
Hence, or otherwise, find the first 3 terms, in ascending powers of $x$, of the expansion of $igg(1 + \frac{x}{2}\bigg)(2 - 3x)^6$
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Answer
Using the result from part (a), we have:
(1+2x)(2−3x)6=(1+2x)(64−576x+2160x2)
Now, we will distribute:
Constant Term: 1⋅64=64
Coefficient of x:
From 1⋅(−576x)=−576x
From 2x⋅64=32x
Therefore, the total is:
−576x+32x=−544x
Coefficient of x2:
From 1⋅2160x2=2160x2
From 2x⋅(−576x)=−288x2
Therefore, the total is:
2160x2−288x2=1872x2
Thus, the first three terms of the expansion are:
64−544x+1872x2
