Question

Can someone please explain how to do this PLEASE!?!?


For the following problems show how the problem is set up, the work to solve the problem, along with the solution.

Expand (x + 3)6 using Pascal’s Triangle. (2 points)

Answer 1

Since exponent is 6, you just need to use Row 6.

Answer 2

But what do I do with (x+3)?

Answer 3

Expand

For example: To expand \((x+2)^2\), you can expand it to \(1\cdot x^22^0 + 2\cdot x^12^1 + 1\cdot x^02^2 \\~\\= x^2+2\cdot2x+2^2\\~\\= x^2+4x+4\)

As we move through each term from left to right, the power of \(x\) decreases from 2 down to zero and power of 2 increase from 0 up to 2.

Then we just put in coefficients of each term using Pascal’s triangle, being with 1, 2, then 1.

Answer 4

Let's work through your problem;
just work with power for now;

That's easy right?
\((x+3)^6 = \text_~x^63^0 + \text_~x^53^1+\text_~x^43^2+\text_~x^33^3+\text_~x^23^4+\text_~x^13^5+\text_~x^03^6 \\~\\~~~~~~~~~~~~~~= \text_~x^6+\text_~3x^5+\text_~9x^4+\text_~27x^3+\text_~81x^2+\text_~243x+\text_~729\)

Now put in number from row 6 of Pascal’s Triangle into blank, as coefficient;

\[\cdots=1\cdot x^6+6\cdot 3x^5+15\cdot 9x^4+ 20\cdot 27x^3+15\cdot 81x^2+6\cdot 243x+1\cdot 729\]

Now multiply coefficients.

Does that make sense?

Answer 5

I think so, soIdentify the coefficients, so first we look for the coefficient then we substitute the second coefficient and the third coefficient. Is that right?

Answer 6

Not sure what you are saying, but you understand the power part, right?

So for these blanks, we just insert numbers from Pascal’s Triangle;

For first term, just put in first number in 6th row;
For second term, just put in second number in same row
For third term, put in third number

And so on...

Answer 7

Yes, now I get it. Thank you so much! :)
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Answer 8

Lol nice ASCII art!