Problem 32 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(3 r^{2}+10 r-8\)

Step-by-Step Solution

Answer

The trinomial factors to \((3r - 2)(r + 4)\).

Step 1: Identify the Trinomial Form

The expression given is a trinomial: \(3r^2 + 10r - 8\) , which is in the form \(ar^2 + br + c\) with \(a = 3\), \(b = 10\), and \(c = -8\).

Step 2: Multiply 'a' and 'c'

Multiply the coefficients 'a' and 'c' to find a number you need to factor. Here, \(3 \times (-8) = -24\).

Step 3: Find Factors of the Product

Determine two numbers that multiply to \(-24\) and add to \(10\) (the 'b' term coefficient). The numbers are \(12\) and \(-2\) since \(12 \times (-2) = -24\) and \(12 + (-2) = 10\).

Step 4: Rewrite the Middle Term

Use the numbers found in Step 3 to split the middle term \(10r\) into two terms: \(12r - 2r\). The expression now is \(3r^2 + 12r - 2r - 8\).

Step 5: Group the Terms

Group terms in pairs: \((3r^2 + 12r) + (-2r - 8)\).

Step 6: Factor Each Group

Factor out the greatest common factor from each group. For \(3r^2 + 12r\), factor out \(3r\), giving \(3r(r + 4)\). For \(-2r - 8\), factor out \(-2\), giving \(-2(r + 4)\).

Step 7: Factor Out the Common Binomial

Notice the common binomial \((r + 4)\) and factor it out: \((3r - 2)(r + 4)\).

Step 8: Check the Solution

Verify the factorization by expanding \((3r - 2)(r + 4)\) to ensure it results in the original trinomial \(3r^2 + 10r - 8\).

Key Concepts

Trinomial

A trinomial is a type of polynomial that contains exactly three terms. In the context of our exercise, the trinomial is expressed as \( 3r^2 + 10r - 8 \). Each term consists of a product of a coefficient and a variable that may be raised to an exponent.
When we talk about trinomials, they are usually represented in the standard form \( ax^2 + bx + c \), where:

• \(a\), \(b\), and \(c\) are constants or coefficients.
• \(ax^2\) is the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.

Understanding the structure of a trinomial is crucial for solving problems because it sets the foundation for factorization. The process involves breaking down a trinomial into simpler components, typically binomials, which we will discuss shortly.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that two or more terms share. In the trinomial factorization process, identifying the GCF becomes crucial during the factoring steps.
In our problem, we identified the GCF while grouping terms in pairs:

• For the pair \( 3r^2 + 12r \), the GCF is \(3r\).
• For the pair \(-2r - 8 \), the GCF is \(-2\).

Factoring out the GCF simplifies the expression and is a stepping stone in turning a complex trinomial into easier-to-handle parts. Taking out the GCF helps reveal common binomials, which lead to the final step of binomial factorization.

Binomial

A binomial is an algebraic expression containing exactly two terms. In our factorization process, the key step was to rewrite the trinomial in such a way that we can factor it into products of binomials.During the exercise, once we factored out the GCF from each grouped pair, we ended up with a common binomial \((r + 4)\). This discovery was pivotal as it allowed us to express the trinomial as a product of two binomials, \((3r - 2)(r + 4)\).
Recognizing the binomial is crucial because:

• It helps simplify complex expressions.
• It makes it easier to solve or further manipulate the expression.

By understanding the structure of binomials, students can confidently approach trinomial factorization, making the overall learning experience more intuitive.

Coefficient

Coefficients are numerical or constant values in front of variables in an algebraic expression. They are vital in understanding and solving polynomial equations, including trinomials.In the given trinomial \( 3r^2 + 10r - 8 \), the coefficients are:

• \(3\) for the quadratic term \(r^2\),
• \(10\) for the linear term \(r\),
• \(-8\) as the constant term.

Coefficients determine the behavior of the terms and their interactions. They are used in operations such as addition, subtraction, and multiplication during the factorization process.
For factorization, manipulating coefficients is essential as seen in our step where \(3\) and \(-8\) were multiplied to guide the factor decomposition process. Recognizing the role of coefficients aids students in understanding the comprehensive layout of polynomial solutions.