Question

Find the coefficient of x^6 in the expansion of (3-x)(2x+1)^10

Answer 1

First consider the expansion of \((2x+1)^{10}\). By the binomial theorem,
\[(2x+1)^{10}=\sum_{k=0}^{10}\binom {10}k(2x)^k1^{n-k}=\sum_{k=0}^{10}\binom {10}k(2x)^k\]
The term containing \(x^5\) and \(x^6\) occur when \(k=5\) and \(k=6\), respectively, so let's grab those terms:
\[\binom {10}5(2x)^5=\cdots\\
\binom{10}6(2x)^6=\cdots\]
We pick these powers specifically because multiplying the \(x^5\) term by the \(-x\) in \(3-x\) generates a term with \(x^6\), while the \(x^6\) term gets multiplied by \(3\) in \(3-x\).

The coefficient of the \(x^6\) term in \((3-x)(2x+1)^{10}\) is then the sum of the coefficients above.

Answer 2

sorry, i'm still a bit confused.

Answer 3

@butterflydreamer
Just to be clear about the question, is it [(3-x)(2x+1)]^10 or (3-x)[(2x+1)^10]

Answer 4

see u will get terms with x^6 in the 5th term and also the 6th term bcoz u gotta multiply it with (3 - x)

Answer 5

the 6th would have x^5 . bt all bcoz u gotta multiply with (3 - x) it would change to x^6

Answer 6

@mathmate the question is:
\[(3-x) (2x+1)^{10}\]

Answer 7

Thank you!
You can continue with @rishavraj or @SithsAndGiggles

Answer 8

@butterflydreamer wht would be the 6th and 5th term??

Answer 9

the 5th term would be......
\[\left(\begin{matrix}10 \\ 5\end{matrix}\right) (2x)^5 \times (3-x) ?\]

Answer 10

yup :))) u see so in this case hmmm the coefficient of x would be -8064 ????

Answer 11

\[\begin{align*}(2x+1)^{10}&=\sum_{k=0}^{10}\binom {10}k(2x)^k\\
&=\cdots+\binom{10}5(2x)^5+\binom{10}6(2x)^6+\cdots\\
&=\cdots+32\binom{10}5x^5+64\binom{10}6x^6+\cdots\end{align*}\]
Multiplying by \(3-x\), you apply the distributive property:
\[(3-x)\left(32\binom{10}5x^5+64\binom{10}6x^6\right)\\
\quad\quad=3\times64\binom{10}6x^6+(-x)\times 32\binom{10}5x^5+\text{other stuff}\\
\quad\quad =3\times64\binom{10}6x^6- 32\binom{10}5x^6+\text{other stuff}\\
\quad\quad =\left(3\times64\binom{10}6- 32\binom{10}5\right)x^6+\text{other stuff}\]

Answer 12

Ohhhhhh right right! I get it now, thank you :))!
I got confused with the expansion :S