Problem 4 - Precalculus

Question

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) \(24 x^{2}+x-3\) (b) \(x^{2}+24 x+144\)

Step-by-Step Solution

Answer

(a) Not irreducible; (b) Not irreducible.

Step 1: Understand the Definition of Irreducibility for Quadratics

A quadratic polynomial \( ax^2 + bx + c \) is irreducible over the real numbers if it has no real roots. This means its discriminant \( b^2 - 4ac \) is less than zero, as a negative discriminant indicates the roots are non-real complex numbers.

Step 2: Apply the Discriminant Formula for Polynomial (a)

Examine the polynomial \( 24x^2 + x - 3 \). Here \( a = 24 \), \( b = 1 \), and \( c = -3 \). Compute the discriminant using formula \( b^2 - 4ac \):\[ Discriminant = 1^2 - 4 \cdot 24 \cdot (-3) = 1 + 288 = 289 \].Since 289 is positive, the roots are real, and therefore the polynomial is not irreducible.

Step 3: Apply the Discriminant Formula for Polynomial (b)

Examine the polynomial \( x^2 + 24x + 144 \). Here \( a = 1 \), \( b = 24 \), and \( c = 144 \). Compute the discriminant:\[ Discriminant = 24^2 - 4 \cdot 1 \cdot 144 = 576 - 576 = 0 \].Since the discriminant is zero, this polynomial has a repeated real root, and thus it is not irreducible either.

Key Concepts

Quadratic Polynomials

Quadratic polynomials are mathematical expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
These expressions form a parabola when plotted on a graph, creating a symmetrical curve around its vertex.Quadratic polynomials are essential in algebra because they commonly appear in various mathematical calculations and real-world applications.
They can represent different scenarios such as projectile motion, area optimization, and economics. When determining the properties of a quadratic polynomial, we look at its coefficients and use several methods for analysis.

• The standard form of a quadratic polynomial: \( ax^2 + bx + c \).
• Vertex form of a quadratic polynomial: \( a(x - h)^2 + k \), where \((h, k)\) is the vertex.
• Factored form highlights the roots of the polynomial.

Each form is useful, depending on what information you need to extract from the expression. Identifying roots helps to determine if a polynomial is irreducible over the real numbers.

Discriminant

The discriminant is a specific value calculated from a quadratic's coefficients \( a \), \( b \), and \( c \), and is given by the formula \( b^2 - 4ac \).
The discriminant helps determine the nature of the roots of a quadratic polynomial without actually solving it.The value of the discriminant tells us whether a quadratic has real or non-real roots, and how many of them:

• If the discriminant is greater than zero, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is less than zero, there are no real roots, meaning the polynomial has complex roots and is irreducible over the real numbers.

In the exercise given, calculating the discriminant for each polynomial helps to determine whether the polynomial is irreducible based on the nature of its roots.

Real Roots

Real roots are the solutions to a polynomial equation that are real numbers. In the context of a quadratic polynomial \( ax^2 + bx + c = 0 \), these are the values of \( x \) for which the equation holds true.
Real roots can be visualized as the points where the polynomial's graph intersects the x-axis.Understanding whether a quadratic has real roots involves exploring its discriminant:

• If there are two distinct real roots, the graph of the quadratic will intersect the x-axis at two points.
• If there is a repeated real root, the graph will just touch the x-axis at one point, known as the vertex of the parabola.

Knowing whether a polynomial has real roots is crucial not only for solving equations but also for assessing particular qualities of graphs. Polynomials determined to be irreducible lack real roots, graphically indicating no intersection with the x-axis and confirming their irreducibility over the real plane.