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

Kaycee Roche

Kaycee Roche

Answered question

2021-08-08

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)=x35x2+4x20

Answer & Explanation

tafzijdeq

tafzijdeq

Skilled2021-08-09Added 92 answers

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)=x35x2+4x20
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)=x35x2+4x20
Rewrite the above polynomial as,
P(x)=x2(x5)+4(x5)
Now, factor the polynomial P(x) as,
P(x)=(x5)(x2+4)
The factor (x2) 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 (x5)(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)=x35x2+4x20
Calculation:
From part (a) the factored form of the polynomial P(x) is,
P(x)=(x5)(x2+4)
Now, factor the remaining quadratic factor to obtain the complete factorization as,
P(x)=(x5)(x2+4)
=(x5)(x2(2i)2)
=(x5)(x2i)(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 (x5)(x2i)(x+2i).

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