Problem 109 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(5 x^{2}-38 x-16\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Quadratic Expression

The given expression is a quadratic polynomial: \(5x^2 - 38x - 16\). Our task is to factor this expression completely.

Step 2: Find Two Numbers for Splitting the Middle Term

We need to find two numbers whose product is equal to the product of the coefficient of \(x^2\) term (which is 5) and the constant term (which is -16), i.e., \(5 \times -16 = -80\), and whose sum is the coefficient of the \(x\) term, i.e., -38.

Step 3: Select the Pair of Numbers

The pair of numbers that multiply to -80 and add to -38 is -40 and 2 because \(-40 \times 2 = -80\) and \(-40 + 2 = -38\).

Step 4: Split the Middle Term

Rewrite the expression by splitting the middle term -38x into -40x and 2x: \(5x^2 - 40x + 2x - 16\).

Step 5: Factor by Grouping

Group the terms into two parts: \((5x^2 - 40x) + (2x - 16)\). Factor out the greatest common factor (GCF) from each group: \(5x(x - 8) + 2(x - 8)\).

Step 6: Factor Out the Common Binomial

Since both groups contain the common binomial factor \((x - 8)\), factor it out: \((5x + 2)(x - 8)\). Thus, the expression is completely factored.

Key Concepts

Polynomial Expressions

A polynomial expression is a mathematical expression that consists of variables and constants. These variables are raised to whole number exponents and are combined using addition, subtraction, and multiplication. A polynomial can have one or more terms. For instance, the expression we are discussing, \(5x^2 - 38x - 16\), is a polynomial. It consists of three terms:

• \(5x^2\), which is the quadratic term
• -38x, which is the linear term
• -16, which is the constant term

Polynomials are quite flexible; they can be as simple as a single term, or they can include multiple terms with varying degrees and coefficients. Understanding the structure of polynomial expressions is important because it is the first step in handling tasks like simplifying or factoring polynomials.

Quadratic Polynomial

A quadratic polynomial is a special type of polynomial that includes a term with the highest power of the variable being 2. The standard form of a quadratic polynomial is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) cannot be zero.
Our expression \(5x^2 - 38x - 16\) fits this mold, making it a quadratic polynomial:

• \(5x^2\) is the quadratic term which makes this expression quadratic
• -38x is the linear term that involves the variable to the first power
• -16 is the constant term and it has no variable associated with it

Quadratic polynomials are common in algebra, and learning to factor them is an essential algebraic skill. Factoring allows us to rewrite quadratic polynomials as products of simpler expressions, which can be helpful in solving equations.

Factoring by Grouping

Factoring by grouping is a technique used to factor certain types of polynomial expressions where traditional factoring methods are not straightforward. This method involves grouping terms with common factors and then factoring separately within those groups.
In our example with \(5x^2 - 38x - 16\), we used factoring by grouping as follows:

• The expression \(-38x\) is split into two terms: \(-40x + 2x\). This simplifies later factoring.
• We then create two groups: \((5x^2 - 40x)\) and \((2x - 16)\).
• Next, we factor each group separately: \(5x(x - 8)\) from the first, and \(2(x - 8)\) from the second.
• The common binomial factor, \((x - 8)\), is then factored out from both groups, giving us the completely factored form: \((5x + 2)(x - 8)\).

Factoring by grouping is particularly useful when dealing with polynomials that do not have a common factor across all terms but can be rearranged into groups that do. This technique can turn complex expressions into manageable products.