## Loujoelou 3 years ago 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.

1. ajprincess

It is nt correct.

2. Loujoelou

Oh okay, can you show me what I did wrong?

3. TuringTest

what is the 5th row of pascal's triangle?

4. Loujoelou

1,4,6,4,1

5. TuringTest

yes, and those will be your coefficients now what are your terms?

6. Loujoelou

x^4 + x^3y + x^2y^2 + xy^3 + y^4

7. TuringTest

no, you have ignored the coefficients in the terms themselves they must remain...

8. Loujoelou

1x^4+4x^3y+6x^2y^2+4xy^3+1y^4 ?

9. TuringTest

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

10. Loujoelou

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?

11. TuringTest

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

12. Loujoelou

I included the 1,4,6,4,1 into the multiplying.

13. TuringTest

let's just look at the terms first without the coefficients then to see where you went wrong

14. Loujoelou

k

15. TuringTest

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

16. Loujoelou

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? :)

17. TuringTest

$\begin{array}91^{st}= (3x)^4(7y)^0=3^4x^4=81x^4\\2^{nd}=(3x)^3(7y)^1=3^3x^3\cdot7^1y^1=567x^3y^1\\3^{rd}=(3x)^2(7y)^2=3^2x^2\cdot7^2y^2=441x^2y^2\\4^{th}=(3x)^1(7y)^3=3x\cdot7^3y^3=1029xy^3\\5^{th}=(3x)^0(7y)^4=7^4y^4=2401y^4\end{array}$yeah but you forgot the y^4 at the end ;)

18. TuringTest

and I wouldn't write them all summed up like that yet until we use the right coefficients from the triangle

19. Loujoelou

oh yeah careless mistake there :) Wait we still have to multiply this with 1,4,6,4,1 right?

20. TuringTest

yes, by the 5th row of the triangle we have the coefficients of the terms as$1(1^{st})+4(2^{nd})+6(3^{rd})+4(4^{th})+1(5^{th})$

21. Loujoelou

K so the final problem would be 81x^4+2268x^3y+2646x^2y^2+4116xy^3+2401 right? :)

22. TuringTest

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 :)

23. TuringTest

besides that everything else is correct though

24. Loujoelou

It should be 756x^3y right?

25. TuringTest

yes :)

26. robtobey

$(3 x+7 y)^4=81 x^4+756 x^3 y+2646 x^2 y^2+4116 x y^3+2401 y^4$

27. TuringTest
28. Loujoelou

Thx a ton @TuringTest for helping me with this problem :)

29. Loujoelou

I know wolframalpha but I wanted to not use anything like that on my test so yeah :) Thx!

30. TuringTest

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.

31. TuringTest

...and you're very welcome!

32. Loujoelou

Oh okay thx :)