Problem 35 - Elementary Algebra

Question

Factor the following problems, if possible. $$ 24 x^{2}-26 x-5 $$

Step-by-Step Solution

Answer

Answer: (6x+1)(4x-5)

Step 1: Identify the expression type

Given expression is quadratic, as it has the form: $$ ax^2 + bx + c $$ Here, a = 24, b = -26, and c = -5.

Step 2: Determine if factoring is possible

To factor the expression, we need to find two numbers that multiply to give ac (product) and add to give b (sum). Product (ac) = 24 * (-5) = -120 Sum (b) = -26 Now, we will look for the factors of -120 that add up to -26.

Step 3: Find the factors

After some trial and error, we can find the pair of factors: -30 and 4. This is because: $$ -30 * 4 = -120 \quad \text{and} \quad -30 + 4 = -26 $$

Step 4: Rewrite middle term and factor by grouping

Rewrite the expression by introducing these two factors instead of the middle term (-26x): $$ 24x^2 -30x + 4x -5 $$ Factor by grouping: $$ 6x(4x -5) +1(4x -5) $$

Step 5: Write the final factored expression

Factor out the common term (4x -5) from both terms of the expression: $$ (6x+1)(4x-5) $$ The factored form of the given quadratic expression is \((6x+1)(4x-5)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomials where the highest power of the variable, usually denoted as \( x \), is 2. These expressions often appear in the format of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the context of the exercise, we see that our expression \( 24x^2 - 26x - 5 \) falls under this category because:

• \( a = 24 \) - the coefficient of \( x^2 \)
• \( b = -26 \) - the coefficient of \( x \)
• \( c = -5 \) - the constant term

Quadratic expressions are common in many areas of mathematics, including algebra and calculus. They can represent parabolic graphs, and solving them is crucial for understanding the intersections of these graphs with the x-axis. Factoring quadratic expressions simplifies them, revealing their roots or x-intercepts.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. With quadratic expressions, such as \( 24x^2 - 26x - 5 \), the aim is to break them down to an expression involving multiplication of two binomials. This factorization aids in solving polynomial equations by setting each factor to zero, simplifying many mathematical analyses.

• Determine the product \( ac = 24 \times (-5) = -120 \).
• Look for two numbers that multiply to \( ac \) and sum to \( b = -26 \).
• Here, the pair \( -30 \) and \( 4 \) satisfies both conditions.

Understanding polynomial factoring is essential as it makes complex expressions more manageable. For quadratic expressions, factoring often reveals important characteristics of their corresponding quadratic equations, such as roots and vertex form.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants. Their main objective is to represent mathematical relationships in a succinct form which can later be manipulated to find unknown values. In this exercise, we focused on simplifying the expression \( 24x^2 - 26x - 5 \).

• The variables in quadratic expressions are raised to a maximum power of 2.
• Brainstorm potential factor pairs for the product of the first and last coefficients, which here equals \( -120 \).
• Verify solution accuracy by expanding the resulting factors to ensure they multiply back to the original expression.

By understanding and manipulating algebraic expressions, students learn how to approach problem-solving in a systematic and logical manner, which is vital for algebra and beyond.

Factoring by Grouping

Factoring by grouping is a method where you segregate terms into groups and factor them individually. This method is especially useful when you need to split a middle term to help factor a quadratic expression. For our problem, the steps involved:

• Rewrite the expression by replacing the middle term \( -26x \) with \( -30x + 4x \).
• Thus, transforming the expression to \( 24x^2 - 30x + 4x - 5 \).
• Group the terms: \((24x^2 - 30x) + (4x - 5)\).
• Factor each group separately to get \( 6x(4x - 5) + 1(4x - 5) \).
• Extract and factor out the common \( (4x-5) \), resulting in \( (6x+1)(4x-5) \).

This technique simplifies complex expressions and is immensely useful in mathematics for breaking down polynomials into simpler, more manageable parts. Once fully mastered, it makes solving and analyzing algebraic expressions much easier.