Problem 33 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ r^{2}-16 r+48 $$

Step-by-Step Solution

Answer

\((r - 12)(r - 4)\)

Step 1: Identify the Greatest Common Factor (GCF)

Examine the terms of the trinomial \( r^2 - 16r + 48 \). The coefficients are 1, -16, and 48. The GCF of these coefficients is 1. Thus, there is no need to factor out any number from the trinomial since the greatest common factor is 1.

Step 2: Break Down the Middle Term

To factor the trinomial \( r^2 - 16r + 48 \), look for two numbers that multiply to the constant term (48) and add up to the coefficient of the middle term (-16). These numbers are -12 and -4, because \(-12 \times -4 = 48\) and \(-12 + -4 = -16\).

Step 3: Factor by Grouping

Rewrite \( r^2 - 16r + 48 \) by breaking down the middle term into two terms using the numbers from Step 2: \( r^2 - 12r - 4r + 48 \). Group the terms: \((r^2 - 12r) + (-4r + 48)\).

Step 4: Factor Each Group

Factor out the greatest common factor from each group. For \(r^2 - 12r\), factor out \(r\), resulting in \(r(r - 12)\). For \(-4r + 48\), factor out \(-4\), resulting in \(-4(r - 12)\).

Step 5: Factor Out the Common Binomial

Notice both groups contain a common binomial \(r - 12\). Factor out \(r - 12\), resulting in \((r - 12)(r - 4)\). This is the completely factored form of the trinomial.

Key Concepts

Greatest Common Factor

The concept of the greatest common factor (GCF) is fundamental in factoring trinomials. It refers to the largest integer or the largest term that evenly divides all the terms within a polynomial. Before diving into more complex factoring techniques, you should always check if there is a common factor across all terms.

In our trinomial, \(r^2 - 16r + 48\), the coefficients are 1, -16, and 48. The GCF of these numbers is 1. Since there is no larger number that can be factored out, we move on to other methods. This step is crucial as it simplifies more complex expressions and makes further factoring easier.

Factoring by Grouping

Factoring by grouping is a powerful technique used to simplify polynomials like trinomials. When you factor by grouping, you split the expression into groups that have common factors. This is particularly useful when direct factoring isn't straightforward.

For \(r^2 - 16r + 48\), we first break the middle term into two parts. We split \(-16r\) into \(-12r\) and \(-4r\), making it possible to group the expression as \((r^2 - 12r) + (-4r + 48)\). Each group contains terms that can be factored further.

• Bigger expressions are broken into simpler parts.
• Look for terms that share common factors.
• Make finding common binomials in expressions more straightforward.

Factoring by grouping helps turn a seemingly complex trinomial into a manageable expression, making it easier to see the factors.

Trinomial

Understanding what a trinomial is helps in solving such polynomial expressions more effectively. A trinomial is a polynomial with exactly three terms. The basic form of a trinomial is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, the trinomial \(r^2 - 16r + 48\) can be factored into two simpler expressions.

Recognizing the general form of a trinomial enables one to decide on suitable factoring techniques. Here:

• The term \(r^2\) stands for the highest degree, signifying it is a quadratic trinomial.
• The \(-16r\) is the middle term, crucial for factoring.
• Constant term \(48\) represents specific value needed to factor the expression entirely.

Understanding these parts is vital when applying techniques like factoring by grouping or identifying special patterns in mathematics.

Factoring Techniques

There is no one-size-fits-all in factoring trinomials, hence understanding different techniques is beneficial. Factoring techniques vary based on the structure and components of the polynomial.

For \(r^2 - 16r + 48\):

• Trial and Error: Find numbers that multiply to 48 (constant term) and sum to -16 (coefficient of middle term).
• Factoring by grouping: Splitting the middle term allows you to group terms and find GCF in smaller parts.

After breaking the middle term, both groups share a common binomial, \(r-12\), which is a key step. Recognizing these patterns helps outline different strategies that lead to the trinomial's complete factorization.