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# I could use the help...

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Use the Binomial Theorem to find the binomial expansion of the given expression.

$$(2x - 3y)^5$$

Jun 6, 2019

#1
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The binomial theorem is as follows:

$$(a+b)^2=\sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^k$$

The formula never helped me much either, but here is the explanation. Looking at any expansion, there are coefficients and exponents. For example, the expansion for $$(a+b)^3$$ is as follows:

$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$

Just looking at the coefficients, you get 1, 3, 3, and 1. This is the fourth row of the Pascal triangle.

For the exponents, you get 3, 2, 1, 0 for a; and 0, 1, 2, 3 for b. From this, we can deduce what $$(2x-3y)^5$$ will be.

The expansion for just $$(x-y)^5$$:

The exponents for x will be 5, 4, 3, 2, 1, 0 and the exponents for y will be 0, 1, 2, 3, 4, 5.

The coefficients will be the sixth row of the Pascal triangle, which is 1, 5, 10, 10, 5, 1

Therefore, the expansion will be $$(x-y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5$$

However, we are not done yet. We need to add the positive/negative signs, alternating.

$$(x-y)^5=x^5-5x^4y+10x^3y^2-10x^2y^3+5xy^4-y^5$$

Using this equation, you can plug in the values to find the given expression.

I hope this helped,

Gavin.

Jun 6, 2019
#2
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expand   (2x - 3y)^5:

32 x^5 - 240 x^4 y + 720 x^3 y^2 - 1080 x^2 y^3 + 810 x y^4 - 243 y^5

Jun 6, 2019