Find a polynomial of the specified degree that satisfies the given conditions.Degree 4; zeros -3, 0, 1, 5; coefficient of x 3 is 6

Question

Find a polynomial of the specified degree that satisfies the given conditions.Degree 4; zeros -3, 0, 1, 5; coefficient of       x    3   is 6

slapadabassyc · Accepted Answer

Step 1Let the polynomial of degree 4 and -3,0,1,5 as zeros be:  f  (  x  )  =  a  [  x  −  (  −  3  )  ]  (  x  −  0  )  (  x  −  1  )  (  x  −  5  )i.e.,   f  (  x  )  =  a  (  x  +  3  )  x  (  x  −  1  )  (  x  −  5  )i.e.   f  (  x  )  =  a  (      x    2    +  3  x  )  (      x    2    −  6  x  +  5  )i.e.,   f  (  x  )  =  a  (      x    4    −  3      x    3    −  13      x    2    +  15  x  )i.e.,   f  (  x  )  =  a      x    4    −  3  a      x    3    −  13  a      x    2    +  15  a  xHere, the coefficient of x3 is 6.Then by condition we have,  −  3  a  =  6i.e.,   a  =      6          −      3      i.e.,   a  =  −  2Step 2Then, the polynomial becomes:  f  (  x  )  =  (  −  2  )  ⋅      x    4    −  3  (  −  2  )  ⋅      x    3    −  13  (  −  2  )  ⋅      x    2    +  15  (  −  2  )  ⋅  xi.e.,   f  (  x  )  =  −  2      x    4    +  6      x    3    +  26      x    2    −  30  xTherefore, the required polynomial is:   f  (  x  )  =  −  2      x    4    +  6      x    3    +  26      x    2    −  30  x

Find a polynomial of the specified degree that satisfies the given conditions. Degree 4; zeros -3, 0, 1, 5; coefficient of x^3 is 6.

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