SOLUTION: using the 6th row of pascals triangle,expand (2x-5y)^5

Question 712684: using the 6th row of pascals triangle,expand (2x-5y)^5
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Pascal's triangle provides coefficients for the terms in the expansion of powers of a binomial (like ). Before we figure out the coefficients we are going to look at the terms.

The pattern of the terms, without the coefficients, for ) will be:

Note how the exponents of each term add up to 5. Note also the alternating operations (positive/add then subtract then add ...). The alternating operations are due to the "-" between the 2x and the 5y in . (If there had been a "+" between the 2x and 5y then there would only be additions.)

We can simplify our coefficient-less expression by raising everything to their powers ...

and then multiplying within the terms...

Now we're ready for the coefficients from Pascal's Triangle. Pascal's Triangle starts with:

                                   1
                                 1   1
                               1   2   1
                             1   3   3   1

As you can see, the outermost edges of the triangle are all 1's. For the numbers on the inside you add the number above and to the left and the number above and to the right. For example, in the 4th row:

The row starts with a 1 because all rows start with a 1.
The first 3 in the 4th row comes from adding the 1 that starts the 3rd row and the 2 in the 3rd row.
The second 3 in the 4th row comes from adding the 2 in the 3rd row and the 1 at the end of the 3rd row.
The row ends with a 1 because all rows end in a 1.

In the 5th row...

It start's with a 1
The next number will be the sum of the first 1 in the 4th row and the first 3 in the 4th row.
Next in the 5th row will be the sum of the two 3's in the 4th row.
Next will be the sum of the second 3 and the last 1
And the row ends with a 1.

I'll leave it up to you to figure out the 5th and 6th rows.

Once you have the 6th row you will have a row with six numbers. Earlier we came up with the expression:

which has six terms! Take each of the numbers from the 6th row, in order, and put one of them in front of each of the 6 terms of the expression above. For example the 1 at the start of the 6th row would go in front of the first term:

The second number in the 6th row works out to be 5. So we put this 5 in front of the second term:
(There will still be a "-" in front of this term.)
The 1 at the end of the 6th row goes in front of the last term:
(There will still be a "-" in front of this term.)
Insert the rest of the 6th row in front of the rest of the terms. And last of all, multiply these numbers from the 6th row with each of the terms. So the first, second and last terms work out to be:

(There will still be a "-" in front of this term.)
(There will still be a "-" in front of this term.)

So you end up with:
+ ? - ? + ?
The three ?'s will be replaced by the terms you figured out. (Make sure you keep the alternating signs!)
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