SOLUTION: Use the Binomial Theorem to expand the binomial. (x+1)^6 show work please.

Question 863468: Use the Binomial Theorem to expand the binomial.
(x+1)^6
show work please.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
First we use Pascal's Triangle to find the binomial coefficients needs. In this case, the outermost exponent is 6. So we look at the row in the triangle that starts with 1, 6, ... (which is highlighted in yellow below)

So the coefficients we'll use are: 1, 6, 15, 20, 15, 6, 1

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The terms of the binomial x+1 are x and 1. We will start off by raising the first term x to the 6th power (the outermost exponent) and 1 to the 0th power to get x^6*1^0 = x^6*1 = x^6.

Then you step down the exponent of 6 to get 6-1 = 5 and at the same time step up the exponent of 0 to 0+1 = 1. The exponent of 5 will be applied to the first term x and the exponent of 1 will be applied to the second term 1. So you'll have x^5*1^1 = x^5*1 = x^5

You keep going: x^4*1^3 = x^4*1 = x^4

and you keep going til the exponent over x reaches 0, so you'll have x^3, x^2, x, 1

Note: The key is that the outermost exponents for each monomial add back up to 6.
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So we got x^6, x^5, x^4, x^3, x^2, x, 1

Finally, we need to multiply each of those terms by the coefficients found in the highlighted row of the pascals triangle above.

So...

1*x^6 = x^6
6*x^5 = 6x^5
15*x^4 = 15x^4
20*x^3 = 20x^3
15*x^2 = 15x^2
6*x = 6x
1*1 = 1

We now have x^6, 6x^5, 15x^4, 20x^3, 15x^2, 6x, 1. Add them all up to get this final answer:

This means is an identity (ie true for all values of x).
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