Anonymous
Anonymous asked in Science & MathematicsMathematics · 8 years ago

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

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  • Mary
    Lv 7
    8 years ago
    Favorite 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.

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