A polynomial P is given (a) Factor P into linear

Tabansi 2021-08-14 Answered

A polynomial P is given (a) Factor P into quadratic factors with real coefficients that are linear and irreducible. (b) Consider only linear components with complex coefficients for P.. 
P(x)=x516x

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Expert Answer

falhiblesw
Answered 2021-08-15 Author has 97 answers

a) To find: The polynomial P(x) is factored by linear, irreducible, quadratic factors with real coefficients.
Given information: 
The polynomial P(x) is, 
P(x)=x516x 
Concept used: 
Linear and Quadratic Factors Theorem: 
The product of linear and irreducible quadratic factors can be factored into any polynomial with real coefficients. 
Calculation: 
The given polynomial P(x) is, 
P(x)=x516x 
Rewrite the above polynomial as, 
P(x)=x(x416) 
=x((x2)242) 
Use identity a2b2=(ab)(a+b) to factor the above equation as, 
P(x)=x(x24)(x2+4) 
=x(x222)(x2+4) 
=x(x2)(x+2)(x2+4) 
The factors x, (x2) and (x+2) are linear factors. 
The factor (x2+4) is irreducible, since it has no real zeros. 
Conclusion: 
Thus, the factored form of the polynomial P(x) that has linear and irreducible quadratic factors is x(x2)(x+2)(x2+4) 
b) To find: The factors of the polynomial P(x) that has linear factors with complex coefficients. 
Given: The polynomial P(x) is, 
P(x)=x516x 
Calculation: 
From part (a) the factored form of the polynomial P(x) is, 
P(x)=x(x2)(x+2)(x2+4) 
Now, factor the remaining quadratic factor to obtain the complete factorization as, 
P(x)=x(x2)(x+2)(x2+4) 
=x(x2)(x+2)(x2(2i)2) 
=x(x2)(x+2)(x2i)(x+2i) 
The above factors are linear factors with complex coefficients. 
Conclusion: 
Thus, the factored form of the polynomial P(x) that has linear factors is x(x2)(x+2)(x2i)(x+2i).

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