Determine whether the given quadratic polynomial is...

Determine whether the given quadratic polynomial is irreducible. [Recall from the text that a quadratic polynomial $f(x)$ is irreducible if the equation $f(x)=0$ has no real roots]
(a) $x^{2}-16$
(b) $x^{2}+16$

Key Concepts

Factorization

Factorization is the method of decomposing a polynomial into a product of lower-degree polynomials. In the context of quadratic polynomials, factorization is directly linked to the existence of real roots. If real roots exist, the quadratic can typically be expressed as a product of two linear factors; otherwise, it is considered irreducible over the reals.

Irreducibility

Irreducibility refers to the property of a polynomial that cannot be factored into simpler polynomials with coefficients in a given field. For a quadratic polynomial over the real numbers, being irreducible means it does not have real roots, and as such, cannot be expressed as a product of two non-constant linear factors.

Real Roots

Real roots are the solutions of the quadratic equation that lie within the set of real numbers. The existence of real roots is critical when assessing the factorability of a quadratic polynomial over the real numbers, as the presence of such roots means it can be factored into linear factors.

Discriminant

The discriminant of a quadratic, calculated as b² - 4ac, is key to determining the nature of its roots. It indicates whether the roots are real and distinct, real and repeated, or complex, thereby influencing the polynomial's factorability over the real numbers.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two, typically written in the form ax² + bx + c, where a, b, and c are real numbers and a is nonzero. It represents one of the simplest non-linear functions and is fundamental in algebra and calculus due to its parabolic graph and well-understood properties.

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