Problem 31 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ x^{2}+13 x+40 $$

Step-by-Step Solution

Answer

Answer: The factored form of the given quadratic trinomial is $$(x + 5)(x + 8)$$.

Step 1: Identify the coefficients and search for the factors

We are given the quadratic trinomial $$x^2 + 13x + 40.$$ In this case, \(b = 13\) and \(c = 40\). We need to find two numbers \(m\) and \(n\) such that \(m \times n = 40\) and \(m + n = 13\).

Step 2: Find the numbers that satisfy the conditions

By examining the factor pairs of 40, we notice that \((m, n) = (5, 8)\) satisfies both conditions: $$5 \times 8 = 40$$ and $$5 + 8 = 13$$.

Step 3: Write the factored form

Now that we have found the values of \(m\) and \(n\), we can write the factored form of the trinomial as $$(x + m)(x + n) = (x + 5)(x + 8)$$.

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra, frequently appearing in various mathematical problems. A quadratic equation is typically expressed in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The highest power of the variable \(x\) within the equation is two, which is why it’s called quadratic. A key aspect of these equations is determining the roots, or solutions, which are the values of \(x\) that make the equation true.

Quadratic equations can often be solved by factoring, which involves rewriting the equation in a product form. This is useful because it’s generally easier to find the roots of a product of expressions. Depending on the equation, factoring might involve grouping terms, using algebraic identities, or making use of patterns like completing a square. However, not all quadratic equations can be factored easily, but when they can, it often reveals neat solution sets.

Factor Pairs

In the process of factoring trinomials, like our example \(x^2 + 13x + 40\), factor pairs are crucial. Factor pairs are simply two numbers that can be multiplied together to produce another number—in this case, the constant term \(c\) of the trinomial.

To factor the trinomial, you need to identify two numbers \(m\) and \(n\) such that:

• \(m \times n = c\), where \(c\) is 40 in our example.
• \(m + n = b\), where \(b\) is 13 in the problem we’re tackling.

To find these factor pairs, you list all pairs of numbers that multiply to the constant term. For 40, these pairs include (1, 40), (2, 20), (4, 10), (5, 8). Among these pairs, only (5, 8) also sum to 13, satisfying both conditions for factoring. Thus, understanding how to manipulate factor pairs is key in simplifying and solving quadratic trinomials.

Trinomial Factoring Technique

The trinomial factoring technique is a strategic method that is particularly useful for expressions of the form \(x^2 + bx + c\). In our exercise, we are dealing with \(x^2 + 13x + 40\), which fits this standard format.

The goal of this technique is to express a quadratic trinomial as a product of two binomials, that is, \((x + m)(x + n)\). Finding the correct binomials requires recognizing the factor pairs that satisfy both conditions previously described: the product \(m \times n = c\) and the sum \(m + n = b\). This involves trial and error, but with practice, recognizing suitable pairs becomes intuitive.

The current trinomial factorization transforms \(x^2 + 13x + 40\) into \((x + 5)(x + 8)\). Once factored, solving the equation when it is set to zero becomes much simpler, as each binomial can individually be set to zero to find the roots \(x = -5\) and \(x = -8\). Mastering this technique is crucial for efficiently solving a broad range of algebraic problems.