Question

Find all zeros of the function and write the polynomial as a product of linear factors

f(x)=x^4+29x^2+100

Answer 1

take x² = t
t²+29t+100
=(t²+25t) +(4t+100)
=t(t+25) + 4(t+25)
=(t+4)(t+25)
=(x²+4)(x²+25)

Answer 2

`f(x)=x^4+29x^2+100=\left(x^2\right)^2+29x^2+100\\\\x^2=v;\ v\geq0\\\\f(v)=v^2+29v+100\\\\a=1;\ b=29;\ c=100\\\\\Delta=b^2-4ac;\ v_1=(-b-\sqrt\Delta)/(2a);\ v_2=(-b+\sqrt\Delta)/(2a)\\\\\Delta=29^2-4\cdot1\cdot100=841-400=441;\ \sqrt\Delta=√(441)=21\\\\v_1=(-29-21)/(2\cdot1)=(-40)/(2)=-20 < 0;\ v_2=(-29+21)/(2\cdot1)=(-8)/(2)=-4 < 0`


`Function\ has\ no\ zeros.\\\\\\f(x)=x^4+29x^2+100=x^4+25x^2+4x^2+100\\\\=x^2(x^2+25)+4(x^2+25)=(x^2+25)(x^2+4)\\\\is\ not\ impossible\ write\ the\ polynomia\l as\ a\ product\ of\ linear\ factors.`