A polynomial f(x) with real coefficients and leading...

A polynomial $f(x)$ with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express $f(x)$ as a product of linear and quadratic polynomials with real coefficients that are irreducible over $\mathbb{R}$.
$$4+3 i,-2+i ; \text { degree } 4$$

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Key Concepts

Complex Conjugate Root Theorem

For any polynomial with real coefficients, if a non-real complex number is a zero, then its complex conjugate must also be a zero. This guarantees that when non-real zeros occur, they come in pairs, thus allowing the construction of quadratic factors with real coefficients.

Polynomial Factorization

This concept involves expressing a polynomial as a product of factors, which may be linear or quadratic, such that when these factors are multiplied together, the original polynomial is recovered. When the polynomial has real coefficients, the factors corresponding to complex zeros are paired as conjugates to yield quadratic polynomials.

Irreducible Quadratic Polynomials Over Real Numbers

A quadratic polynomial with real coefficients is said to be irreducible over the reals if it has no real roots, meaning its discriminant is negative. In the context of polynomials with complex zeros, the paired complex conjugate roots form quadratic factors that cannot be factored further into real linear factors.

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