If m-nx +28xsquare + 12xcube+9xpower4 is a perfect square, find the values of m and n

Question

If m-nx +28xsquare + 12xcube+9xpower4 is a perfect square, find the values of m and n

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1 month 2021-08-13T01:52:40+00:00 1 Answer 0 views 0

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    2021-08-13T01:54:21+00:00

    Answer:

    Step-by-step explanation:

    It is given that the polynomial m−nx+28x  

    2

    +12x  

    3

    +9x  

    4

     is a perfect square, we must equate it to the square of general form of equation that is (ax  

    2

    +bx+c)  

    2

     as shown below:

    9x  

    4

    +12x  

    3

    +28x  

    2

    −nx+m=(ax  

    2

    +bx+c)  

    2

     

    ⇒9x  

    4

    +12x  

    3

    +28x  

    2

    −nx+m=(ax  

    2

    )  

    2

    +(bx)  

    2

    +(c)  

    2

    +(2×ax  

    2

    ×bx)+(2×bx×c)+(2×c×ax  

    2

    )

    (∵(a+b+c)  

    2

    =a  

    2

    +b  

    2

    +c  

    2

    +2ab+2bc+2ca)

    ⇒9x  

    4

    +12x  

    3

    +28x  

    2

    −nx+m=a  

    2

    x  

    4

    +b  

    2

    x  

    2

    +c  

    2

    +2abx  

    3

    +2bcx+2acx  

    2

     

    Now, comparing the coefficients, we get:

    a  

    2

    =9,b  

    2

    +2ac=28,c  

    2

    =m,2ab=12,2bc=−n

    a  

    2

    =9

    ⇒a=3

    2ab=12

    ⇒2×3×b=12

    ⇒6b=12

    ⇒b=2

    b  

    2

    +2ac=28

    ⇒2  

    2

    +(2×3c)=28

    ⇒4+6c=28

    ⇒6c=28−4

    ⇒6c=24

    ⇒c=4

    c  

    2

    =m

    ⇒m=4  

    2

     

    ⇒m=16

    2bc=−n

    ⇒n=−2bc

    ⇒n=−2×2×4

    ⇒n=−16

    Hence, m=16 and n=−16.

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