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Math Help - Binomial Expansions, what is the term

  1. #1
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    Smile Binomial Expansions, what is the term

    Hello!
    This is the question,
    find the term x^3 in

    Code:
    (3x +4 ) (x − 2)^4

    I dont know how to start or what is ment by the 'term',

    do i expand (x − 2)^4?



    Thank you!!
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  2. #2
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    Quote Originally Posted by appleseed View Post
    Hello!
    This is the question,
    find the term x^3 in

    Code:
    (3x +4 ) (x − 2)^4
    I dont know how to start or what is ment by the 'term',

    do i expand (x − 2)^4?



    Thank you!!
    Expand (x-2)^4and multiply it with (3x+4) and collect the terms containing x^3.
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  3. #3
    MHF Contributor
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    West Malaysia
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    Quote Originally Posted by appleseed View Post
    Hello!
    This is the question,
    find the term x^3 in

    Code:
    (3x +4 ) (x − 2)^4

    I dont know how to start or what is ment by the 'term',

    do i expand (x − 2)^4?



    Thank you!!
    hi

    yes you expand but not everything ,

    (3x+4)(x-2)^4=(3x+4)(x^4+4(-2)x^3+6(-2)^2x^2+...)

    =(6x^2)(4)(3x)+(4)(-2)(4x^3)

    notice that you can ignore those terms which will not generate x^3 when multiplied .
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  4. #4
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    Thank you guys! this really helps
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  5. #5
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    Since you titled this "binomial expansions", presumably you know the "binomial coefficients" and using those you don't need to do all that work.

    You can get x^3 by multiplying that "3x" by the x^2 term in the fourth power expansion and by multiplying "4" by the x^3 term.

    The x^2 term in (x- 2)^4 is \begin{pmatrix}4 \\ 2\end{pmatrix}(-2)^2x^2 and the x^3 term is \begin{pmatrix}4 \\ 3\end{pmatrix}(-2)^1x^3.

    Multiplying the first by 3x and the second by 4 gives \left(\begin{pmatrix}4 \\ 2\end{pmatrix}(3)(-2)^2+ \begin{pmatrix}4 \\ 3\end{pmatrix}(4)(-2)\right)x^3
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