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Loujoelou
Explain, in complete sentences, how you would expand (3x + 7y)^4 using Pascal’s Triangle. I got 3x^4+84x^3y+126x^2y^2+84xy^3+7y^4. Can anyone verify to see if this is correct? :) Thank you very much.
Oh okay, can you show me what I did wrong?
what is the 5th row of pascal's triangle?
yes, and those will be your coefficients now what are your terms?
x^4 + x^3y + x^2y^2 + xy^3 + y^4
no, you have ignored the coefficients in the terms themselves they must remain...
1x^4+4x^3y+6x^2y^2+4xy^3+1y^4 ?
the terms are 3x and 7y so you have 1 4 6 4 1 1(3x)^4(7y)^0+4(3x)^3(7y)^1+6(3x)^2(7y)^2+4(3x)^1(7y)^3+1(3x)^0(7y)^4
When I combine everything I still get what I still got from up above. o.O 1st term- 3x^4 2nd term- 84x^3y 3rd term- 126x^2y^2 4th term- 84xy^3 5th term= 7y^4 Am I combining some of these terms wrong?
did you try to include the coefficients from pascals triangle yet or no? because what you have is not quite right any way I can slice it, but it's closer
I included the 1,4,6,4,1 into the multiplying.
let's just look at the terms first without the coefficients then to see where you went wrong
terms:\[\begin{array}91^{st}= (3x)^4(7y)^0=3^4x^4=81x^4\\2^{nd}=(3x)^3(7y)^1=3^3x^3\cdot7^1y^1=567x^3y^1\\3^{rd}=(3x)^2(7y)^2\\4^{th}=(3x)^1(7y)^3\\5^{th}=(3x)^0(7y)^4\end{array}\]you failed to distribute the exponents properly... I will continue but this should be enough to see your mistake we then put in the coefficients from Pascal's triangle at the end and add it all up
Oh I forgot to use the exponent on the numbers too. So it should actually be- 81x^4+567x^3y^1+441x^2y^2+1029x^1y^3+2401 correct? :)
\[\begin{array}91^{st}= (3x)^4(7y)^0=3^4x^4=81x^4\\2^{nd}=(3x)^3(7y)^1=3^3x^3\cdot7^1y^1=567x^3y^1\\3^{rd}=(3x)^2(7y)^2=3^2x^2\cdot7^2y^2=441x^2y^2\\4^{th}=(3x)^1(7y)^3=3x\cdot7^3y^3=1029xy^3\\5^{th}=(3x)^0(7y)^4=7^4y^4=2401y^4\end{array}\]yeah but you forgot the y^4 at the end ;)
and I wouldn't write them all summed up like that yet until we use the right coefficients from the triangle
oh yeah careless mistake there :) Wait we still have to multiply this with 1,4,6,4,1 right?
yes, by the 5th row of the triangle we have the coefficients of the terms as\[1(1^{st})+4(2^{nd})+6(3^{rd})+4(4^{th})+1(5^{th})\]
K so the final problem would be 81x^4+2268x^3y+2646x^2y^2+4116xy^3+2401 right? :)
the second coefficient is wrong, but that is due to a typo I made above I can see find it and fix it! that should be your effort in this :)
besides that everything else is correct though
It should be 756x^3y right?
\[(3 x+7 y)^4=81 x^4+756 x^3 y+2646 x^2 y^2+4116 x y^3+2401 y^4 \]
http://www.wolframalpha.com/input/?i=expand%20(3x%20%2B%207y)%5E4%20
Thx a ton @TuringTest for helping me with this problem :)
I know wolframalpha but I wanted to not use anything like that on my test so yeah :) Thx!
Yeah I just gave it for you to check your work. You should never rely on it; it is wrong more often than you would probably think at first.
...and you're very welcome!