Problem 25 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$3 n^{2}-7 n-20$$

Step-by-Step Solution

Answer

The polynomial factors as \((3n + 5)(n - 4)\).

Step 1: Identify the Polynomial

We start with the polynomial expression given: \(3n^{2} - 7n - 20\). Our goal is to factor this quadratic polynomial completely. The polynomial is of the form \(ax^2 + bx + c\), where \(a = 3\), \(b = -7\), and \(c = -20\).

Step 2: Multiply 'a' and 'c'

Multiply the coefficient of \(n^2\), which is \(a = 3\), by the constant term \(c = -20\). This gives us \(3 \times -20 = -60\). We need to find two numbers that multiply to \(-60\) and add to \(-7\), the middle term coefficient.

Step 3: Determine the Pair that Satisfies Conditions

We look for two numbers whose product is \(-60\) and whose sum is \(-7\). The numbers \(-12\) and \(5\) satisfy these conditions because \(-12 \times 5 = -60\) and \(-12 + 5 = -7\).

Step 4: Decompose the Middle Term

Use the numbers obtained in Step 3 to break down the middle term \(-7n\) into \(-12n + 5n\). So, \(3n^{2} - 7n - 20\) becomes \(3n^{2} - 12n + 5n - 20\).

Step 5: Factor by Grouping

Group the terms: \((3n^{2} - 12n) + (5n - 20)\). Now factor out the common factors in each group: \(3n(n - 4) + 5(n - 4)\).

Step 6: Factor out the Common Binomial

Notice both groups contain the binomial factor \((n - 4)\). Factor \((n - 4)\) out: \((3n + 5)(n - 4)\). The polynomial is fully factored now.

Key Concepts

Quadratic Polynomials

Understanding quadratic polynomials is essential before diving into factoring techniques. A quadratic polynomial is a polynomial of degree 2, which means the highest power of the variable (commonly denoted as \(x\) or \(n\)) is 2. It has the general form \(ax^2 + bx + c\), where:

• \(a\), the coefficient of \(x^2\), is non-zero.
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

Quadratic polynomials appear frequently in various mathematical problems. They create parabolic curves when graphed and have specific methods for solutions, such as factoring, completing the square, or using the quadratic formula.
Understanding the structure and properties of these polynomials is key to factor them effectively into simpler expressions, or to identify if they further solveable by another method.

Factor by Grouping

Factor by grouping is an effective method for factorizing certain polynomials, including quadratic polynomials, like the one we are dealing with: \(3n^{2} - 7n - 20\). By grouping, we split the polynomial into terms that can share common factors. The strategy involves the following steps:

• Identify two pairs of terms from the polynomial that make sense to group together. Often, the middle term is decomposed using previously found numbers that multiply to \(ac\) and add to \(b\) (in our case, \(-12\) and \(5\) for \(-7\)).
• Apply the grouping: write the polynomial as the sum of two pairs of products, \((3n^2 - 12n) + (5n - 20)\).
• Factor out the greatest common factor in each group: \(3n(n - 4) + 5(n - 4)\).

This approach simplifies the polynomial into a form where a common binomial factor (\((n - 4)\) in this case) can be factored out, leading to the solution. The consistent result indicates the polynomial is fully factored when no further common factors are apparent.

Polynomial Expressions

Polynomials are algebraic expressions that consist of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents. A polynomial expression is considered fully factored when it is written as a product of its simplest expressions.

• Quadratic polynomials are a subset of polynomial expressions.
• The factoring process helps transform a polynomial into a more manageable form, usually involving simpler binomials or monomials.
• Understanding how polynomial expressions function lays the foundation for solving equations and modeling real-world situations.

Recognizing and factoring polynomial expressions is crucial in algebra and calculus, working through complex calculations by breaking them down into simpler steps. Properly factoring polynomials can make understanding and solving equations more efficient, as illustrated by factoring \(3n^2 - 7n - 20\) into \((3n + 5)(n - 4)\).
This approach also reveals potential roots of the polynomial, which tell us the points where the corresponding graph intersects the horizontal axis.