How do you find a polynomial function that has zeros #x=-4, -1, 3, 6# and degree n=4?

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Answer 1 · 2017-01-25T18:53:39

The Reqd. Poly. Fun. is

#k(x^4-4x^3-23x^2+54x+2), k in RR-{0}.#

Let us denote by #p(x)# the reqd. poly. fun.

#x=-4" is a zero of "p(x):. (x-(-4))=(x+4)# is a factor.

On the similar lines, #(x+1), (x-3) and (x-6)# are also factors.

As the degree of #p(x)# is #4,# #p(x)# can not have any more

factors, except some constant, say, #k!=0#.

Accordingly, we have,

#p(x)=k(x+4)(x+1)(x-3)(x-6)#

#=k{(x+4)(x-6)}(x+1)(x-3)#

#=k{(x^2-2x-24)(x^2-2x-3)}#

#=k(y-24)(y-3), [y=x^2-2x]#

#=k(y^2-27y+72)#

#=k{(x^2-2x)^2-27(x^2-2x)+72}#

#=k{x^4-4x^3+4x^2-27x^2+54x+72}#

#:. p(x)=k(x^4-4x^3-23x^2+54x+2), k in RR-{0},# is the reqd. poly.

Enjoy Maths.!