Problem 131 - Introductory Algebra for College Students

Question

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. The factorable trinomial \(4 x^{2}+8 x+3\) and the prime trinomial \(4 x^{2}+8 x+1\) are in the form \(a x^{2}+b x+c\) but \(b^{2}-4 a c\) is a perfect square only in the case of the factorable trinomial.

Step-by-Step Solution

Answer

The statement 'The factorable trinomial \(4x^{2}+8x+3\) and the prime trinomial \(4x^{2}+8x+1\) are in the form \(ax^{2}+bx+c\) but \(b^{2}-4ac\) is a perfect square only in the case of the factorable trinomial.' makes sense. This is because, the discriminant of the first trinomial is a perfect square, making it factorable and the discriminant of the second trinomial is not a perfect square, making it prime.

Step 1 Title

Determine the value of \(a\), \(b\), and \(c\) for each trinomial. For the trinomial \(4x^{2}+8x+3\), \(a=4\), \(b=8\), and \(c=3\). For the trinomial \(4x^{2}+8x+1\), \(a=4\), \(b=8\), and \(c=1\).

Step 2 Title

Calculate the discriminant \(b^{2}-4ac\) for both trinomials. The value of the discriminant for the first trinomial is \(64 - 4*4*3 = 16\), while for the second trinomial is \(64 - 4*4*1 = 48\)

Step 3 Title

Check if the discriminants are perfect squares. A perfect square is a number that can be expressed as the product of an integer with itself. 16 is a perfect square because it can be expressed as \(4^2\), but 48 is not since it cannot be expressed as the product of an integer with itself.

Step 4 Title

Draw a conclusion based on the results. Since the discriminant for the first trinomial is a perfect square, this trinomial is factorable. However, the discriminant for the second trinomial is not a perfect square, hence this trinomial is prime.

Key Concepts

Factorable Trinomials

A trinomial is an algebraic expression with three terms, usually in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Factorable trinomials are those trinomials which can be rewritten as the product of two binomials. This is done through a process known as factoring.

To determine if a trinomial is factorable, we often use the discriminant, which is given by the formula \(b^2 - 4ac\). If the discriminant is a perfect square, then the trinomial is factorable. For instance, with the trinomial \(4x^2+8x+3\), we identify \(a=4\), \(b=8\), and \(c=3\) and calculate the discriminant to be 16, which is a perfect square (\(4^2\)). Therefore, this trinomial can be factored into the product of binomials.

Mastery of recognizing factorable trinomials is crucial for simplifying algebraic expressions and solving quadratic equations, which often form the foundation for more complex mathematical concepts.

Prime Trinomials

Contrary to factorable trinomials, prime trinomials, also known as irreducible trinomials, cannot be factored into the product of two binomial expressions. These trinomials have no factors other than 1 and the trinomial itself. The discriminant is key in identifying such trinomials.

For example, consider the trinomial \(4x^2+8x+1\). Using the values \(a=4\), \(b=8\), and \(c=1\), we compute the discriminant to be 48. Because 48 is not a perfect square, the trinomial is prime and does not factor neatly into simpler expressions. The study of prime trinomials can be particularly important when dealing with higher-level algebra and number theory, as it leads to a deeper understanding of algebraic structures and their properties.

Perfect Squares

The term perfect square refers to a number that is the square of an integer. In other words, a perfect square can be written as \(n^2\), where \(n\) is an integer. In the context of quadratic equations and functions, recognizing perfect squares is essential for factoring trinomials and solving equations.

Continuing with our previous examples, the discriminant of the first trinomial (16) is indeed a perfect square since it can be expressed as \(4^2\). This not only verifies that the trinomial is factorable, but also simplifies the process of finding its roots. However, the second discriminant (48) is not a perfect square, indicating that the corresponding trinomial is prime.

Understanding perfect squares assists in various areas of mathematics, from geometry to calculus, as it helps simplify radical expressions, find the length of sides in right triangles, and much more. Recognizing and using perfect squares are fundamental skills that serve as building blocks for advanced mathematical concepts.