Consider the polynomial f(X)=x^4+1 a) Explain why f has no real roots, and why this means f mustfactor as a product of two ireducible quadratics. b) Factor f and find all of its complex roots.

postillan4 2021-01-05 Answered
Consider the polynomial f(X)=x4+1
a) Explain why f has no real roots, and why this means f mustfactor as a product of two ireducible quadratics.
b) Factor f and find all of its complex roots.
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Expert Answer

Ezra Herbert
Answered 2021-01-06 Author has 99 answers

a) For the roots,
f(x)=0
x4+1=0
x4=1
But there is no real number ehose even power gives negative real number which means f has no real roots.
Futher since f has no real roots so f does not have any linear factors.
Therefore f must factor as a product of two irreducible qudratics.
b) f(x)=x4+1
x1+2x22x2 Adding and substracting 2x2
=(x2+1)2(2)2
=(x2+1+2x)(x2+12x)a2=b2=(a+b)(ab)
=(x2+2x+1)(x22x+1)
For the roots.
f(x)=0
(x2+2x+1)(x22x+1)=0
(x2+2x+1)orx22x+1=0
x=(2±(2)24(1)(1)2(1),x=(2±(2)24(1)(1)2(1)
x=2±242,x=2±242
x=2±22,x=2±22
x=2±i22,x=2±i22
x=1±i2,x=1±i2
Therefore the comlex roots are,
1+i2,1i2,1+i2,1i2

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