Problem 74 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ 8 m^{2}+5 m-10 $$

Step-by-Step Solution

Answer

The factored form is \((8m - 5)(m + 2)\).

Step 1: Identify the Quadratic Expression

The expression given is a quadratic trinomial: \(8m^2 + 5m - 10\). This is in the form \(ax^2 + bx + c\) where \(a = 8\), \(b = 5\), and \(c = -10\).

Step 2: Check for Common Factors

First, check if there is a greatest common factor (GCF) for the terms 8, 5, and -10. There isn't any common factor other than 1, so there's no simplification needed at this step.

Step 3: Apply the AC Method

The AC method involves multiplying \(a\) and \(c\): \(8 \times (-10) = -80\). We need two numbers that multiply to \(-80\) and add to \(5\) (the coefficient of \(m\)).

Step 4: Find the Pair of Numbers

Possible pairs that multiply to \(-80\) include (1, -80), (2, -40), (4, -20), (5, -16), (8, -10). The pair that sums to 5 is \(16\) and \(-5\).

Step 5: Rewrite the Expression

Rewrite the middle term using the pair: \(8m^2 + 16m - 5m - 10\). This breaks the expression to help with factoring.

Step 6: Factor by Grouping

Group the terms to factor: \((8m^2 + 16m) + (-5m - 10)\).Factor each group separately:1. \(8m(m + 2)\)2. \(-5(m + 2)\)The expression becomes \(8m(m + 2) - 5(m + 2)\).

Step 7: Factor the Common Binomial

Since \((m + 2)\) is common in both terms:\((8m - 5)(m + 2)\).

Step 8: Verify Solution

Expand \((8m - 5)(m + 2)\) to ensure it equals the original expression:\(8m^2 + 16m - 5m - 10\).Combine like terms:\(8m^2 + 11m - 10\), which matches the rearranged terms.

Key Concepts

Factoring Trinomials

Factoring trinomials is a fundamental skill in algebra that simplifies complex expressions, solving equations efficiently. It involves breaking down a complex quadratic equation into a product of simpler factors.Understanding the structure of the quadratic trinomial is crucial. This structure is \( ax^2 + bx + c \), where "a," "b," and "c" are coefficients. These variables must be carefully handled to successfully rewrite the expression in a factored form.Here’s the approach when factoring:

• Identify the quadratic form and its coefficients—"a," "b," and "c".
• Determine if there's any greatest common factor (GCF) that can be factored out initially.
• If no GCF exists, proceed to other factoring methods, such as the AC method, which is often needed here.

Mastering this strategy provides a solid foundation in algebra, preparing you for more complicated polynomial manipulations.

AC Method

The AC method is a reliable technique to factor quadratic trinomials, especially when conventional factoring isn't straightforward. This method involves breaking down and rewriting the middle term to simplify factoring.Here’s how it works:

• Multiply coefficients "a" and "c" from the trinomial \( ax^2 + bx + c \). The result helps find two critical numbers.
• Search for a pair of numbers that multiply to the product \( ac \) yet add up to "b"—the coefficient of the linear term.
• Split the middle term following these numbers to rewrite the trinomial.

For example, in the expression \( 8m^2 + 5m - 10 \), multiplying \( a \, \text{and} \, c \) gives \(-80\), and finding numbers like \(16\) and \(-5\) leads to a handy formation for subsequent grouping.Once refactored, it subsequently simplifies the process of factor grouping, making it both a strategic and effective approach in polynomial factoring.

Polynomials

Polynomials are algebraic expressions that form the foundation of numerous mathematical concepts. Understanding polynomials, especially in their simplest forms like monomials, binomials, and trinomials, is key to manipulating more complex equations.Polynomials like \( 8m^2 + 5m - 10 \) typically involve powers of variables, with coefficients attached to introduce variability. These terms play different roles depending on their degree— the highest power in the equation.The key to efficiently working with polynomials includes:

• Recognizing the structure and form—knowing whether an expression is a simple monomial or a more complex trinomial.
• Managing coefficients and variables with apt calculation methods.
• Applying the right factoring techniques, helping simplify and solve polynomial equations.

Whether used for graphing functions or solving real-world problems, polynomials are a versatile and fundamental aspect of algebra that once mastered, open doors to advanced mathematics.