Problem 59 - Beginning and Intermediate Algebra

Question

Factor completely. $$13 t^{2}+17 t-18$$

Step-by-Step Solution

Answer

The completely factored expression of \(13t^2 + 17t - 18\) is \((t + 14)(13t - 1)\).

Step 1: Identify the coefficient of each term:

In this quadratic expression, the coefficient of the \(t^2\) term is \(13\), the coefficient of the \(t\) term is \( 17\), and the constant term is \(-18\).

Step 2: Determine the product of the first and last coefficients:

Next, multiply the coefficient of the \(t^2\) term by the constant term: \(13 \times -18 = -234\).

Step 3: Find two numbers whose product is equal to \( -234\) and sum is equal to \(17\):

After performing trial and error or using an algorithm, we find that the pair of numbers is \((18, -13)\) because \(18 \times -13 = -234\) and \(18 + -13 = 17\).

Step 4: Rewrite the middle term using the two numbers:

Now, rewrite the expression by breaking up the \(17t\) term into the sum of \(18t\) and \(-13t\): \(13t^2 + 18t - 13t - 18\).

Step 5: Group the terms and use the distributive property:

Next, group the terms as follows: \((13t^2 + 18t) - (13t + 18)\). Apply the distributive property by factoring out the greatest common factor (GCF) for each pair of terms: \(13t(t + 14) - 1(t + 14)\).

Step 6: Factor by grouping:

Since both terms have a common binomial factor of \((t+14)\), we can factor that out, giving us the result: \((t+14)(13t-1)\). So the completely factored expression of \(13t^2 + 17t - 18\) is \((t + 14)(13t - 1)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are a type of algebraic expression where the highest degree of its variable is 2. These types of expressions are commonly found in various algebraic contexts and can take various forms. The general format of a quadratic expression is \( ax^2 + bx + c \), where:

• \( a \) is the coefficient of the term \( x^2 \)
• \( b \) is the coefficient of the linear term \( x \)
• \( c \) is the constant term

Quadratic expressions often appear in problems related to physics, engineering, and statistics. Solving these expressions can help predict trajectories and compute areas, just to name a few applications. Understanding how to work with these expressions is a key skill in algebra, especially when factoring is involved.

Factoring by Grouping

Factoring by grouping is a powerful strategy used to break down more complex expressions into simpler ones. It is especially useful when dealing with polynomials that do not easily factor into smaller terms. The process starts by identifying pairs of terms that can be grouped and factored separately.

To effectively factor by grouping, you should:

• Reorganize the polynomial if necessary to identify common factors between pairs
• Identify the greatest common factor (GCF) in each pair
• Apply the distributive property to factor out the GCF from each pair
• Recognize any common binomial factors that can be factored out

This method is useful when the polynomial has four terms, as seen in the example provided. Factoring by grouping simplifies the process of breaking down quadratic expressions into simpler products.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions can represent a wide range of real-world and mathematical scenarios.

Key components of algebraic expressions include:

• Terms: Each part of the expression separated by a plus or minus sign
• Coefficients: Numbers that multiply the variables within the terms
• Constants: Terms without any variables
• Operators: The arithmetic symbols that show how terms are combined

In algebra, expressions represent a form of shorthand to describe patterns and solve equations or inequalities. Mastering how to manipulate and factor these expressions is essential for progressing in mathematics. Different factoring techniques, like factoring by grouping, can be applied depending on the complexity and structure of the expression.

Polynomials

Polynomials are fundamental components in algebra, consisting of terms that are combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can have constants, variables, and the exponents that tell us the degree of each term.

Important features of polynomials include:

• The degree: This is the highest exponent of the variable, which indicates the polynomial's order
• Coefficients: Numbers attached to the variables
• Constant terms: They are plain numbers without any variables attached

Factoring polynomials is crucial because it can reveal roots and simplify complex expressions for easier problem-solving. Techniques like factoring by grouping are particularly useful when dealing with higher-degree polynomials or those that show specific patterns in their terms. Knowing how to factor effectively is an invaluable skill for algebra students working with these versatile mathematical expressions.