How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are -2, -4, -7?

↳Redirected from "Suppose that I don't have a formula for #g(x)# but I know that #g(1) = 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
1 Answer
AA_123
Dec 5, 2016

The zeros are: #color(red)(-2),color(blue)(-4)# and #color(green)(-7)#

Find the factors of the function in the form of #(x+z)#

The first factor:-
#zcolor(red)(-2)=0#
#z=2#

#(x+2)#

The second factor:-
#zcolor(blue)(-4)=0#
#z=4#

#(x+4)#

The third factor:-
#zcolor(green)(-7)=0#
#z=7#

#(x+7)#

Multiply the factors to get the least-degree function

#(x+2)(x+4)(x+7)=(x^2+4x+2x+8)(x+7)#

#=(x^2+6x+8)(x+7)#

#=x^3+7x^2+6x^2+42x+8x+56#

Combine like terms

#f(x)=x^3+13x^2+50x+56#

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