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+8x2−9

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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+8x2−9

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a) To find: The factorization of the polynomial P(x) in linear and irreducible quadratic factors with real coefficients.  Given information:  The polynomial P(x) is,  P(x)=x4+8x2−9  Concept used:  Linear and Quadratic Factors Theorem:  Any polynomial that has real coefficients can be factored into the product of linear and irreducible quadratic factors.  Calculation:  The given polynomial P(x) is,  P(x)=x4+8x2−9  Rewrite the above polynomial as,  P(x)=x4+9x2−x2−9  Now, factor the polynomial P(x) as,  P(x)=x2(x2+9)−1(x2+9)  =(x2−1)(x2+9)  =(x−1)(x+1)(x2+9)  The factor (x−1) and (x+1) are linear factors with real zeros.  The factor (x2+9) 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−1)(x+1)(x2+9)  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)=x4+8x2−9  Calculation:  From part (a) the factored form of the polynomial P(x) is,  P(x)=(x−1)(x+1)(x2+9)  Now, factor the remaining quadratic factor to obtain the complete factorization as,  P(x)=(x−1)(x+1)(x2+9)  =(x−1)(x+1)(x2−(3i)2)  =(x−1)(x+1)(x−3i)(x+3i)  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−1)(x+1)(x−3i)(x+3i).

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

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