Question

A polynomial function that has roots 1 and 3 i. The complex conjugates theorem states that must also be a root.

Answer 1

Complex conjugates are two complex numbers and so must occur in pairs

• That is if there is 3 + i
• There must also be 3 - i
• X=(3+i) X=(3-i)

The conjugate root theorem tell us t that when the complex number a + bi is a root of a polynomial P(x) in one variable with real coefficients, then the complex conjugate a - bi is also a root of that polynomial.

Complex conjugates are often referred to as two complex numbers, that is they have 2 forms a + bi,

• a = real part of a complex numbers
• i = √ -1.
• bi = imaginary part of complex number

The two complex numbers are often referred to as conjugates of each other. That is they will have the same real part and their imaginary parts are negatives of each other.

Example are:

• a+ bi is a - bi.
• 5 + 3i is 5 - 3i

Conclusively we can therefore say that getting the complex conjugate of a complex number entails changing the sign of the imaginary part of the number.

• That is if there is 3 + i
• There must also be 3 - i

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Answer 2

Done

A polynomial function that has roots 1 and 3 + i.

The complex conjugates theorem states that  

✔ 3 – i

must also be a root.

The factors of the polynomial are