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# TRIGNOMETRY QUESTION. 10 POINTS FOR ALL U GENIUSES?

What is the numerical coefficient of the term containing x^3y^2 in the expansion of (x+2y)^5

anddd

What is the middle term in the expansion of (x^2+1/x)^6

### 1 Answer

- MaryLv 78 years agoFavorite Answer
1.

(x + 2y)^5

= (x + 2y)[(x + 2y)^2]^2

= (x + 2y)(x^2 + 4xy + 4y^2)^2

= x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5

x^2 4xy 4y^2

x^2 4xy 4y^2 (*

------- ------ -------

x^4. 4x^3y.. 4x^2y^2

........ 4......... 16..................16xy^3

........ ............. 4..................16............ 16y^4

------- ------------- ----------------- ---------- --------

x^4.. 8x^3y.. 24x^2y^2...... 32xy^3......16y^4

x...... 2y (*

------------

x^5... 8x^4y.. 24x^3y^2.. 32x^2y^3.. 16xy^4

........ 2.......... 16............. 48.............. 64......... 32y^5

--------- ------------ --------- --------- ------------- --------- ---

x^5 10x^4y.. 40x^3y^2... 80x^2y^3... 80xy^4.. 32y^5

The numerical coefficient of the term containing x^3y^2 is 40.

2.

(x^2+1/x)^6

= (x^2+1/x)^2*[(x^2+1/x)^2]^2

= (x^2+1/x)^2*[x^4 + 2x + x^(-2)]^2

= (x^2+1/x)^2*[x^8 + 2x^5 + 6x^2 + 4x^(-1) + x^(-4)]

= x^12 + 4x^9 + 11x^6 + 28x^3 + 13 + 6x^(-3) + x^(-6)

Note 1/x = x^(-1)

x^2........ x^(-1)

x^2........ x^(-1) (*

----- ------ --------

x^4........ x

........ ..... x........ x^(-2)

------- --------- -----------

x^4....... 2x....... x^(-2)

x^4....... 2x....... x^(-2) (*

------- --------- -------- ----

x^8....... 2x^5....1x^2

............ 2......... 4.......... 2x^(-1)

............ ........... 1.......... 2......... x^(-4)

------ -------- -------- ------- ------- ----- ----

x^8...... 2x^5.... 6x^2 ....4x^(-1). x^(-4)

x^4...... 2x........ x^(-2) (*

------ --------- ------ -------

x^12.... 2x^9.... 6x^6.... 4x^3.....1x^0

........... 2.......... 4.........12.......... 8..... 2x^(-3)

........... ............ 1.......... 2.......... 6..... 4.......... 1x^(-6)

------- ------- ------ --------- -------- --------- ---------- ------ -

x^12... 4x^9....11x^6... 28x^3...13..... 6x^(-3)... x^(-6)

The middle term in the expansion is 28x^3.

Note that I used a long hand multiplication. This is my own method. I found it easier on the computer to calculate from left to right.