A polynomial P is given. a) Factor P into linear and irreducible quadratic factor with real coefficients. b) Factor P completely into linear factors with complex coefficients. P(x)=x3−5x2+4x−20

Question

A polynomial P is given.  a) Factor P into linear and irreducible quadratic factor with real coefficients.  b) Factor P completely into linear factors with complex coefficients.  P(x)=x3−5x2+4x−20

tafzijdeq · Accepted Answer

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)=x3−5x2+4x−20  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)=x3−5x2+4x−20  Rewrite the above polynomial as,  P(x)=x2(x−5)+4(x−5)  Now, factor the polynomial P(x) as,  P(x)=(x−5)(x2+4)  The factor (x−2) is a linear factor.  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−5)(x2+4).  b) To find: The factors of the polynomial P(x) that has linear factors with complex coefficients.  The polynomial P(x) is,  P(x)=x3−5x2+4x−20  Calculation:  From part (a) the factored form of the polynomial P(x) is,  P(x)=(x−5)(x2+4)  Now, factor the remaining quadratic factor to obtain the complete factorization as,  P(x)=(x−5)(x2+4)  =(x−5)(x2−(2i)2)  =(x−5)(x−2i)(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−5)(x−2i)(x+2i).

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

Answered question

Answer & Explanation

New Questions in Pre-Algebra