How can I simplify this binomial completely using pascal's triangle?

Question

I'm in honor's Algebra 2 and before christmahaunakwanzica break, I was assigned a project that I know forget how to do. We have to expand a binomial using Pascal's Triangle. My problem is (5x-3y)^9. So far I got

1(5x)^9 + 9(5x)^8(-3y)^1 + 36(5x)^7(-3y)^2 + 84(5x)^6(-3y)^3 + 126(5x)^5(-3y)^4 + 126(5x)^4(-3y)^5 + 84(5x)^3(-3y)^6 + 36(5x)^2(-3y)^7 + 9(5x)^1(-3y)^8 + (-3y)^9 .

Now the problem is I don't know what to do know to simplify it more. I'm senile now when it comes to math because I haven't been to school for so long. I hope someone can help cause this is due tomorrow...

Answer 1

Your expansion is good. Now all you have to do is to distribute the exponents.

Recall the property: (ab)^n = (a^n)(b^n)

Term 1:
1(5x)^9 =1(5^9)(x^9) = 1953125(x^9)

Term 2:
9(5x)^8(-3y)^1 = 9(5^8)(x^8)((-3)^1)(y^1) = -10546875(x^8)y

Term 3:
36(5x)^7(-3y)^2 = 36(5^7)(x^7)((-3)^2)(y^2) = 25312500(x^7)(y^2)

Term 4:
84(5x)^6(-3y)^3 = 84(5^6)(x^6)((-3)^3)(y^3) = -35437500(x^6)(y^3)

Term 5:
126(5x)^5(-3y)^4 = 126(5^5)(x^5)((-3)^4)(y^4) = 31893750(x^5)(y^4)

etc...

Do this for all 10 terms and then add them together to get your final "simplified" answer.

Good luck!

Answer 2

alsh is right. There is one thing I can suggest. Google binomial expansion to see what it brings up. Some binomial expansions can be summed by special series, like the Taylor Series. To check to see that you expanded this correctly, and to me it looks correct, let x = y = 1. Then the entire expansion collapses to (5-3)^9 =(2)^9 = 512.

Answer 3

Pascals triangle is extremely easy as quickly as you get the cling of what happening: 2^6 + 6(2)^5(sqrt(5)) + 15(2)^4(sqrt(5))^2 + 20(2)^3(sqrt(5))^3 + 15(2)^2(sqrt(5))^4 + 6(2)(sqrt(5))^5 + (sqrt(5))^6 next you will possibly desire to multiply it out: =sixty 4+ 192(sqrt(5)) + 1200 + 800(sqrt(5)) + 1500 + 3 hundred(sqrt(5)) + a hundred twenty five =2889+1292(sqrt(5)) =5778 wish that facilitates, Ptr

Answer 4

(5x-3y)^9 = (summation i=0 to 9)( (9,i) * [(5x)^(9-i)] * (3y)^i )