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)=x3−2x−4

Question

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)=x3−2x−4

Jaylen Fountain · Accepted Answer

a) To find: The factorization of the polynomial P(x) in linear and irreducible quadratic factors with real coefficients. Consider, P(x)=x3−2x−4 Using the Rational Zeros Theorem, possible rational zeros are ±1,±2,±4. Substitute x=2∈P(x)=x3−2x−4, we get P(2)=23−2(2)−4 ⇒P(2)=8−4−4 ⇒P(2)=0 Therefore, x=2 is a factor of P(x)=x3−2x−4. Factorize the polynomial, P(x)=x3−2x−4. ⇒P(x)=(x−2)(x2+2x+2) Therefore, P(x)=(x−2)(x2+2x+2). b) To find: The factor completely into linear factors with complex coefficients. Consider, P(x)=x3−2x−4 Using the Rational Zeros Theorem, possible rational zeros are ±1,±2,±4. Substitute x=2∈P(x)=x3−2x−4, we get P(2)=23−2(2)−4 ⇒P(2)=8−4−4 ⇒P(2)=0 Therefore, x=2 is a factor of P(x)=x3−2x−4. Factorize the polynomial, P(x)=x3−2x−4 ⇒P(x)=(x−2)(x2+2x+2) ⇒P(x)=(x−2)(x−(−1+i))(x−(−1−i)) Therefore, P(x)=(x−2)(x−(−1+i))(x−(−1−i)).

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

Answered question

Answer & Explanation

New Questions in Pre-Algebra