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)=x6−64

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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)=x6−64

SoosteethicU · Accepted Answer

a) To find: The factor of the polynomial into linear and irreducible quadratic factor with real coefficients.  Consider, P(x)=x6−64  Using the Rational Zeros Theorem, possible rational zeros are ±1,±2,±4,±8,±16,±32,±64.  Substitute x=2∈P(x)=x6−64, we get  P(2)=26−64  ⇒P(2)=64−64  ⇒P(2)=0  Therefore, x=2 is a factor of P(x)=x6−64.  Factorize the polynomial, P(x)=x6−64  ⇒P(x)=(x−2)(x5+2x4+4x3+8x2+16x+32)  Substitute x=−2∈(x−2)(x5+2x4+4x3+8x2+16x+32), we get  ((−2)5+2(−2)4+4(−2)3+8(−2)2+16(−2)+32)  =−32+32−32+32−32+32  =0  Therefore, x=−2 is a factor of (x5+2x4+4x3+8x2+16x+32).  Factorize the polynomial, (x5+2x4+4x3+8x2+16x+32)  ⇒(x5+2x4+4x3+8x2+16x+32)=(x+2)(x4+4x2+16)  ⇒P(x)=(x−2)(x+2)(x4+4x2+16)  ⇒P(x)=(x−2)(x+2)[

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

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