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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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.

Ezra Herbert · Accepted Answer

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+2x2−2x2 Adding and substracting 2x2 =(x2+1)2−(2)2 =(x2+1+2x)(x2+1−2x)∵a2=b2=(a+b)(a−b) =(x2+2x+1)(x2−2x+1) For the roots. f(x)=0 ⇒(x2+2x+1)(x2−2x+1)=0 ⇒(x2+2x+1)orx2−2x+1=0 ⇒x=(−2±(2)2−4(1)(1)2(1),x=−(−2±(2)2−4(1)(1)2(1) ⇒x=−2±2−42,x=2±2−42 ⇒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,−1−i2,1+i2,1−i2

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.

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