Problem 28 - Elementary and Intermediate Algebra

Question

Factor each trinomial. See Example 4 or Example \(10 .\) $$a^{2}-10 a-39$$

Step-by-Step Solution

Answer

The factorization of \(a^2 - 10a - 39\) is \((a - 13)(a + 3)\).

Step 1: Identify the Trinomial

The given trinomial is \( a^2 - 10a - 39 \). This is a quadratic trinomial in standard form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -10 \), and \( c = -39 \).

Step 2: Find Two Numbers

We need to find two numbers that multiply to \(-39\) (the constant term \(c\)) and add to \(-10\) (the coefficient of the linear term \(b\)). The numbers \(-13\) and \(3\) satisfy these conditions: \(-13 \times 3 = -39\) and \(-13 + 3 = -10\).

Step 3: Write the Middle Term as a Sum

Rewrite the middle term \(-10a\) as \(-13a + 3a\). This gives us \(a^2 - 13a + 3a - 39\).

Step 4: Group and Factor by Grouping

Group the terms into two pairs: \((a^2 - 13a)\) and \((3a - 39)\). Factor out the greatest common factor in each group: \(a(a - 13) + 3(a - 13)\).

Step 5: Factor Out the Common Binomial

Notice that \((a - 13)\) is a common factor in the expression. Factor it out: \((a - 13)(a + 3)\).

Step 6: Verify the Factorization

Expand \((a - 13)(a + 3)\) to ensure it equals the original trinomial \(a^2 - 10a - 39\): \(a^2 + 3a - 13a - 39 = a^2 - 10a - 39\). The factorization is correct.

Key Concepts

Quadratic Equations

Quadratic equations are a cornerstone of algebra, appearing frequently in both mathematics and real-world applications. A quadratic equation is one where the highest power of the variable is a square. The general form for a quadratic equation is \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).

Quadratic equations are significant because they describe parabolic paths, commonly seen in physics problems involving projectiles. Understanding the structure of these equations helps in predicting the shape and position of the parabola.

• **Coefficient \( a \):** Determines the direction and width of the parabola. If \( a > 0 \), the parabola faces upward; if \( a < 0 \), it faces downward.
• **Coefficient \( b \):** Influences the placement of the vertex, affecting the symmetry of the parabola.
• **Constant \( c \):** Represents the y-intercept, indicating where the parabola crosses the y-axis.

Recognizing a quadratic equation like \( a^2 - 10a - 39 \) enables you to apply various methods to solve it, with factoring being one of the most powerful techniques.

Polynomial Factoring

Polynomial factoring is a method used to express polynomials as a product of simpler polynomials. This process is vital because it simplifies expressions and solves polynomial equations effectively. Factoring is like the reverse of expanding polynomials; while multiplying terms to get a polynomial can be straightforward, reversing the process requires understanding patterns and techniques.

In the given example \( a^2 - 10a - 39 \), the goal is to express it as \((a - 13)(a + 3)\). Here's how you can achieve that:

• **Identify the trinomial:** Look for patterns in the quadratic expression.
• **Find numbers that multiply to \( c \) and add to \( b \):** This is the key step in selecting the correct pair of binomials.
• **Break the middle term:** Rewrite the quadratic expression by breaking the middle term based on the numbers found.
• **Factor by grouping:** Organize the terms into pairs and factor out common terms to simplify the expression.

The objective is often simplifying polynomial equations to make solving more intuitive, akin to cracking a code with logical steps.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. They form the building blocks of algebra, encapsulating relationships and operations in a compact format.

Expressions can denote anything from simple calculations, like \( 2x + 5 \), to more complex ones like \( a^2 - 10a - 39 \). The power of algebraic expressions lies in their ability to symbolize problems abstractly, which can then be manipulated using various mathematical operations.

• **Variables:** They are symbols that represent numbers, making equations flexible and generalizable.
• **Coefficients:** These numbers multiply the variables, indicating their magnitude and affect within the expression.
• **Constants:** Numbers on their own, which remain the same throughout any evaluation or transformation of the expression.

Manipulating algebraic expressions, such as factoring trinomials, enhances skills in problem-solving and logical reasoning. Representing a complicated expression in a simpler form, you can easily interpret and apply it across various mathematical problems.