Answer:
C. [tex](5-\frac{1}{2})^6[/tex]
Step-by-step explanation:
Given
[tex]15(5)^2(-\frac{1}{2})^4[/tex]
Required
Determine which binomial expansion it came from
The first step is to add the powers of he expression in brackets;
[tex]Sum = 2 + 4[/tex]
[tex]Sum = 6[/tex]
Each term of a binomial expansion are always of the form:
[tex](a+b)^n = ......+ ^nC_ra^{n-r}b^r+.......[/tex]
Where n = the sum above
[tex]n = 6[/tex]
Compare [tex]15(5)^2(-\frac{1}{2})^4[/tex] to the above general form of binomial expansion
[tex](a+b)^n = ......+15(5)^2(-\frac{1}{2})^4+.......[/tex]
Substitute 6 for n
[tex](a+b)^6 = ......+15(5)^2(-\frac{1}{2})^4+.......[/tex]
[Next is to solve for a and b]
From the above expression, the power of (5) is 2
Express 2 as 6 - 4
[tex](a+b)^6 = ......+15(5)^{6-4}(-\frac{1}{2})^4+.......[/tex]
By direct comparison of
[tex](a+b)^n = ......+ ^nC_ra^{n-r}b^r+.......[/tex]
and
[tex](a+b)^6 = ......+15(5)^{6-4}(-\frac{1}{2})^4+.......[/tex]
We have;
[tex]^nC_ra^{n-r}b^r= 15(5)^{6-4}(-\frac{1}{2})^4[/tex]
Further comparison gives
[tex]^nC_r = 15[/tex]
[tex]a^{n-r} =(5)^{6-4}[/tex]
[tex]b^r= (-\frac{1}{2})^4[/tex]
[Solving for a]
By direct comparison of [tex]a^{n-r} =(5)^{6-4}[/tex]
[tex]a = 5[/tex]
[tex]n = 6[/tex]
[tex]r = 4[/tex]
[Solving for b]
By direct comparison of [tex]b^r= (-\frac{1}{2})^4[/tex]
[tex]r = 4[/tex]
[tex]b = \frac{-1}{2}[/tex]
Substitute values for a, b, n and r in
[tex](a+b)^n = ......+ ^nC_ra^{n-r}b^r+.......[/tex]
[tex](5+\frac{-1}{2})^6 = ......+ ^6C_4(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+ ^6C_4(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
Solve for [tex]^6C_4[/tex]
[tex](5-\frac{1}{2})^6 = ......+ \frac{6!}{(6-4)!4!)}*(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+ \frac{6!}{2!!4!}*(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+ \frac{6*5*4!}{2*1*!4!}*(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+ \frac{6*5}{2*1}*(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+ \frac{30}{2}*(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+15*(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+15(5)^{6-4}(\frac{-1}{2})^4+.......[/tex]
[tex](5-\frac{1}{2})^6 = ......+15(5)^2(\frac{-1}{2})^4+.......[/tex]
Check the list of options for the expression on the left hand side
The correct answer is [tex](5-\frac{1}{2})^6[/tex]
