Linear and Quadratic Factors. A polynomial P is given (a) Factor P into linear and irreducible quadratic factors with real coefficients. (b) Factor P completely into linear factors with complex coefficients. P(x)=x4+8x2+16

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Linear and Quadratic Factors. A polynomial P is given (a) Factor P into linear and irreducible quadratic factors with real coefficients. (b) Factor P completely into linear factors with complex coefficients.  P(x)=x4+8x2+16

tabuordg · Accepted Answer

a) To find: The factor of the polynomial into linear and irreducible quadratic factor with real coefficients.  Consider, P(x)=x4+8x2+16  ⇒P(x)=x4+4x2+4x2+16  ⇒P(x)=x2(x2+4)+4(x2+4)  ⇒P(x)=(x2+4)(x2+4)  Therefore, P(x)=(x2+4)(x2+4).  b) To find: The factor completely into linear factors with complex coefficients.  Consider, P(x)=x4+8x2+16  ⇒P(x)=x4+4x2+4x2+16  ⇒P(x)=x2(x2+4)+4(x2+4)  ⇒P(x)=(x2+4)(x2+4)  ⇒P(x)=(x−2i)(x−2i)(x+2i)(x−2i)  Therefore, P(x)=(x−2i)(x−2i)(x+2i)(x−2i).

Linear and Quadratic Factors. A polynomial P is given (a)

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