Problem 63 - Beginning and Intermediate Algebra

Question

Will the sum of two trinomials always be a trinomial? Why or why not? Give an example.

Step-by-Step Solution

Answer

Yes, the sum of two trinomials will always be a trinomial. When adding two trinomials in the form \(ax^2 + bx + c\) and \(dx^2 + ex + f\), their sum will have the form \((a + d)x^2 + (b + e)x + (c + f)\), which is still a trinomial. For example, adding the trinomials \(2x^2 + 3x + 4\) and \(5x^2 - x + 1\) results in \(7x^2 + 2x + 5\), which is also a trinomial.

Step 1: Definition of a Trinomial

A trinomial is an algebraic expression consisting of three terms. Typically, a trinomial can be written in the form \(ax^2 + bx + c\), where a, b, and c are constants.

Step 2: Adding Two Trinomials

To determine whether the sum of two trinomials will always be a trinomial, we can analyze the addition of two trinomials in general form. Consider two trinomials: \(ax^2 + bx + c\) and \(dx^2 + ex + f\).

Step 3: Sum of Trinomials

Now, let's add these two trinomials together: \((ax^2 + bx + c) + (dx^2 + ex + f) = (a + d)x^2 + (b + e)x + (c + f)\) The sum is still an algebraic expression with three terms, meaning it remains a trinomial.

Step 4: Example of Adding Two Trinomials

Let's consider the trinomials \(2x^2 + 3x + 4\) and \(5x^2 - x + 1\). When we add them together, we get: \((2x^2 + 3x + 4) + (5x^2 - x + 1) = (2 + 5)x^2 + (3 - 1)x + (4 + 1) = 7x^2 + 2x + 5\) The sum remains a trinomial, which supports our initial analysis. Therefore, the sum of two trinomials will always be a trinomial.