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

facas9

facas9

Answered question

2021-08-15

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)=x32x4

Answer & Explanation

Jaylen Fountain

Jaylen Fountain

Skilled2021-08-16Added 169 answers

a) To find: The factorization of the polynomial P(x) in linear and irreducible quadratic factors with real coefficients.
Consider, P(x)=x32x4
Using the Rational Zeros Theorem, possible rational zeros are ±1,±2,±4.
Substitute x=2P(x)=x32x4, we get
P(2)=232(2)4
P(2)=844
P(2)=0
Therefore, x=2 is a factor of P(x)=x32x4.
Factorize the polynomial, P(x)=x32x4.
P(x)=(x2)(x2+2x+2)
Therefore, P(x)=(x2)(x2+2x+2).
b) To find: The factor completely into linear factors with complex coefficients.
Consider, P(x)=x32x4
Using the Rational Zeros Theorem, possible rational zeros are ±1,±2,±4.
Substitute x=2P(x)=x32x4, we get
P(2)=232(2)4
P(2)=844
P(2)=0
Therefore, x=2 is a factor of P(x)=x32x4.
Factorize the polynomial, P(x)=x32x4
P(x)=(x2)(x2+2x+2)
P(x)=(x2)(x(1+i))(x(1i))
Therefore, P(x)=(x2)(x(1+i))(x(1i)).

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