polynomials (1 Viewer)

[bos1234]

rt3 + i is one root of x^4 + cx^2 + e = 0, where c and e are real. Find c and e, and factor x^4 + cx^2 + e into quadratic factors with real coefficients

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Prove that if two polynomials P(x) and Q(x) have a common factor (x-a) then (x-a) is also a factor of P(x) - Q(x).

 

[onebytwo]

rt3 + i is one root of x^4 + cx^2 + e = 0, where c and e are real. Find c and e, and factor x^4 + cx^2 + e into quadratic factors with real coefficients

put rt3+i in the equation given, do the same for rt3-i, since the coefficients are real the conjugate is also a root. then using simult equations find c and e...then division? im not sure

 

[jyu]

Prove that if two polynomials P(x) and Q(x) have a common factor (x-a) then (x-a) is also a factor of P(x) - Q(x).

P(x) = (x - a)U(x)
Q(x) = (x - a)V(x)

P(x) - Q(x) = (x - a)U(x) - (x - a)V(x)
= (x - a)[U(x) - V(x)]

.: x - a is a factor of P(x) - Q(x).

:wave:

 

[AlvinCY]

For the first question, this is my guess:

rt3 + i is one root of x⁴ + cx² + e = 0, where c and e are real. Find c and e, and factor x⁴ + cx² + e into quadratic factors with real coefficients.

I’m assuming rt3 means root 3, and I’m going to use ./ for a root.

./3 + i is a root, therefore
(./3 + i)⁴ + c(./3 + i)² + e = 0

9 + 4(./3)³i + 6(./3)²i² + 4(./3)i³ + 1(i⁴) + c (3 + 2./3 i – 1) + e = 0
9 + 12./3 i – 18 – 4./3 i + 1 + 3c + 2./3 c i – c + e = 0
- 8 + 2c + e + 2./3 i (4 + c) = 0

Clearly by equating coefficients, two complex numbers are only equal if and only if their real and imaginary parts equate.

- 8 + 2c + e = 0, and 2./3 (4 + c) = 0 meaning that c = -4
Substituting into - 8 + 2c + e = 0; - 8 – 8 + e = 0; e = 16

So c = -4 and e = 16?

As for the factorising part x4 - 4x² + 16 can be written as (x - ./3 – i)(x - ./3 + i)(x – a)(x – b) where and b are the other two roots we want to find.

(x - ./3 – i)(x - ./3 + i)(x – a)(x – b)
= [(x - ./3)² + 1](x – a)(x – b)
= (x² - 2./3 x + 4)(x – a)(x – b)

So, (x² - 2./3 x + 4) is a factor… carry out polynomial division, you’ll find that: x⁴ - 4x² + 16 = (x² - 2./3 x + 4) (x² + 2./3 x + 4)