Problem 59 - Elementary and Intermediate Algebra

Question

Write each trinomial in descending powers of one variable and factor. See Example 8 . $$r^{3}-16 r+6 r^{2}$$

Step-by-Step Solution

Answer

The trinomial factors to \(r(r + 8)(r - 2)\).

Step 1: Rearrange the Trinomial

First, rewrite the trinomial \(r^3 - 16r + 6r^2\) in descending order of the powers of \(r\). This means arranging the terms from the highest power to the lowest power. The correct order is \(r^3 + 6r^2 - 16r\).

Step 2: Factor Common Term

Look for any common terms in the expression \(r^3 + 6r^2 - 16r\). We can factor out an \(r\) from each term, giving us \(r(r^2 + 6r - 16)\).

Step 3: Factor the Quadratic

Now, factor the quadratic \(r^2 + 6r - 16\). We need two numbers that multiply to \(-16\) and add to \(6\). These numbers are \(8\) and \(-2\). Thus, we can write \(r^2 + 6r - 16\) as \((r + 8)(r - 2)\).

Step 4: Combine All Factors

Combine the factored quadratic with the factored out \(r\) to write the complete factorization of the original trinomial. The final factored form is \(r(r + 8)(r - 2)\).

Key Concepts

Understanding Polynomial Order

Polynomials are expressions consisting of variables raised to various powers and coefficients. To properly work with them, especially in tasks like factoring, it's important to understand their order. The order, or degree, of a polynomial is determined by the highest power of the variable present. For example, in the expression \(r^3 + 6r^2 - 16r\), the term \(r^3\) has the highest power, making the order of the polynomial 3. Reordering polynomials from the highest to the lowest degree helps in systematic analysis and simplification. This way, factoring and other operations become more manageable.

The Basics of Common Term Factoring

Common term factoring is a handy technique to simplify polynomials. This process involves identifying terms within a polynomial that share a common factor, such as a variable or number. Once identified, you factor out the common factor from all terms, which can simplify the polynomial significantly.

For instance, in \(r^3 + 6r^2 - 16r\), every term contains the variable \(r\). By factoring \(r\) out, the polynomial simplifies to \(r(r^2 + 6r - 16)\). This step is crucial because it often makes subsequent factoring of the polynomial easier, leading to the complete factorization of the expression.

Quadratic Factoring Made Easy

Factoring a quadratic expression involves breaking it down into two simpler binomials. These quadratics take the form \(ax^2 + bx + c\). The key to quadratic factoring is finding two numbers that multiply to \(c\) and add to \(b\).

In the quadratic \(r^2 + 6r - 16\), the task is to identify numbers that multiply to \(-16\) and add up to \(6\). Here, these numbers are \(8\) and \(-2\). Thus, the quadratic factors into \((r + 8)(r - 2)\). This method is crucial for simplifying complex expressions and solving quadratic equations.

The Importance of Mathematics Education

Mathematics education empowers students with the skills necessary to manipulate and understand various mathematical concepts. The ability to factor expressions like trinomials is not just about following steps; it's about understanding relationships and developing problem-solving skills.

By engaging with these concepts, students enhance their critical thinking, helping them in various fields beyond mathematics. A strong foundation in mathematics opens doors to scientific disciplines, technology roles, and even logical reasoning required in everyday decision-making. Emphasizing understanding over rote memorization makes mathematics education a powerful tool for lifelong learning.