Problem 93 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because some trinomials are prime, some quadratic equations cannot be solved by factoring.

Step-by-Step Solution

Answer

The statement makes sense. Some trinomials are prime and cannot be factored. Therefore, some quadratic equations, which are a type of trinomial, cannot be solved by factoring.

Step 1: Understanding the terms

Trinomials are algebraic expressions composed of three unlike terms, while Quadratic equations are equations of the second order (The highest exponent is 2). A Prime trinomial is a trinomial that cannot be factored.

Step 2: Analyzing the statement

The given statement says - Because some trinomials are prime, some quadratic equations cannot be solved by factoring. To understand the validity of this statement, there needs to be a grasp of the relationship between trinomials and quadratic equations. The standard form of a quadratic function is \( ax^2 + bx + c \), which is a trinomial. Since prime trinomials cannot be factored, if a quadratic equation (which is a form of trinomial) is a prime trinomial, it, therefore, cannot be factored.

Step 3: Conclusion

From the analysis, it can be deduced that the statement makes sense. This is because there are indeed some quadratic equations that are prime trinomials and therefore cannot be factored, making it impossible to solve such quadratic equations by factoring.