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Math Help - Another Polynomial question

  1. #1
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    Another Polynomial question

    The curve y=(ax+b)^2(c-x) cuts the x-axis at 3 and touches the x axis at x=9/4. Find the values of a,b,c.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by requal View Post
    The curve y=(ax+b)^2(c-x) cuts the x-axis at 3 and touches the x axis at x=9/4. Find the values of a,b,c.
    If one point is on the x-axis, then y=0. So the point (3,0) is a point on the curve.

    If it "touches" the x axis at x=9/4, it means that the x axis is tangent to the curve at this point.
    This means two things :
    - the derivative at this point is 0.
    - the point (9/4,0) is a point on a curve.

    Now, substitute all this information in the equation. It should give you 3 equations with 3 unknowns.
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  3. #3
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    Quote Originally Posted by Moo View Post
    Hello,

    If one point is on the x-axis, then y=0. So the point (3,0) is a point on the curve.

    If it "touches" the x axis at x=9/4, it means that the x axis is tangent to the curve at this point.
    This means two things :
    - the derivative at this point is 0. Mr F adds: If you haven't done calculus, this means that there's a turning point at this point.

    - the point (9/4,0) is a point on a curve.

    Mr F adds: In case you haven't done calculus ..... when a cubic function has a turning point at the x-intercept ( {\color{red}\alpha}, 0) this means that {\color{red}(x - \alpha)^2} is a factor of the cubic.

    Now, substitute all this information in the equation. It should give you 3 equations with 3 unknowns.
    ..
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  4. #4
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    Could you also take advantage of the fact that where it "touches" the x axis the power of the bracket must be even?

    eg.



    Since it cuts the x axis the power of the bracket must be odd (in this case 1):



    I haven't worked through the rest of it yet but will this way work as well?

    (you would need to find another equation linking and c and b or c and a. Differentiation can be used for this).
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