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Learn the definition, examples and proof of the theorem that states that if a + bi is a root of a real polynomial, then so is a − bi. Find out how this theorem relates to odd-degree polynomials and eigenvalues of real matrices.
Learn what is a complex conjugate of a complex number, how to find it, and its properties. Explore the complex conjugate root theorem, the product of a complex number and its conjugate, and the complex conjugate of a matrix.
Learn the definition, properties and uses of the complex conjugate of a complex number, which is the number with the same real part and opposite imaginary part. Find out how the complex conjugate relates to polynomials, functions, matrices and vector spaces.
Complex Conjugate Root Theorem states that for a real coefficient polynomial \(P(x)\), if \(a+bi\) (where \(i\) is the imaginary unit) is a root of \(P(x)\), then so is \(a-bi\) To prove this, we need some lemma first.
The complex conjugate of a complex number is formed by changing the sign between the real and imaginary components of the complex number. Given a complex number of the form, z = a + b i. where a is the real component and b i is the imaginary component, the complex conjugate, z*, of z is: z* = a - b i.
In algebra, the Complex Conjugate Root Theorem states that if is a polynomial with real coefficients, then a complex number is a root of if and only if its complex conjugate is also a root.
Complex Conjugates - Arithmetic. Given a complex number z = a + bi \, (a, b \in \mathbb {R}) z = a+ bi(a,b ∈ R), the complex conjugate of z, z, denoted \overline {z}, z, is the complex number \overline {z} = a - bi z = a−bi. The complex conjugate has the same real component a a, but has opposite sign for the imaginary component b b.
