Problem 32 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. \(r^{2}-15 r-16\)

Step-by-Step Solution

Answer

The factorization of the trinomial \(r^{2}-15 r-16\) is \((r - 16)(r + 1)\)

Step 1: Identify the trinomial

The trinomial given in this exercise is \(r^{2}-15 r-16\)

Step 2: Factor the trinomial

In order to factor the trinomial, look for two numbers that add to -15 (the coefficient of r) and multiply to -16 (the constant term). The numbers that satisfy those conditions are -16 and 1. Therefore, the factorization of the trinomial is \((r - 16)(r + 1)\) .

Step 3: Check the factorization using the FOIL method

The FOIL method involves four multiplications: \(r \cdot r\) gives \(r^{2}\), \(r \cdot 1\) gives \(r\), \(-16 \cdot r\) gives \(-16r\), and \(-16 \cdot 1\) gives \(-16\). Adding them, \(r^{2} + r - 16r - 16 = r^{2}- 15r - 16\), which matches the original trinomial, confirming the correct factorization.

Key Concepts

Algebra

Algebra is a fundamental area of mathematics that deals with symbols and the rules for manipulating these symbols. It serves as the unifying thread of almost all mathematics, making it crucial for understanding various mathematical concepts. At its core, algebra involves working with equations, expressions, and knowing how to manipulate these to find unknown values. When working with algebra, especially with polynomials like trinomials, there are standard operations and methods to follow for solving equations. This includes techniques such as

• Factoring, which means expressing a number or an algebraic expression as a product of its factors.
• Simplifying expressions, which involves rewriting them in the simplest form possible, making calculations easier.

Grasping the basics of algebra opens up a wide range of possibilities for problem-solving and is particularly helpful when approaching polynomial expressions.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. One of the simplest forms of polynomials is a trinomial, which is an expression composed of three terms. For example, the polynomial given in the problem is a trinomial: \[ r^{2} - 15r - 16 \] Understanding the structure of polynomials is essential for performing operations like factoring.Factoring polynomials relies on breaking down the expression into simpler components that can be multiplied to get the original polynomial, in this case, \( (r - 16)(r + 1) \). A factorization is correct if, after factoring, the expression can be expanded using methods like FOIL to return to the original polynomial expression.Recognizing the coefficients and constant terms is crucial when attempting to find factors of a polynomial. In trinomials like the one presented, you focus on two main tasks:

• Finding two numbers that multiply to the constant term.
• Finding two numbers that add or subtract to yield the middle term.

This systematic approach ensures you factor the polynomial correctly.

FOIL Method

The FOIL method is a key technique used in algebra to multiply two binomials. FOIL stands for First, Outer, Inner, Last, describing the order in which you multiply the terms of each binomial.To multiply \( (r - 16)(r + 1) \), use the FOIL steps:

• First: Multiply the first terms of each binomial: \(r imes r = r^{2} \).
• Outer: Multiply the outer terms: \(r imes 1 = r \).
• Inner: Multiply the inner terms: \(-16 imes r = -16r \).
• Last: Multiply the last terms: \(-16 imes 1 = -16 \).

After calculating these products, you add them together to form the expression \( r^{2} + r - 16r - 16 \), which simplifies to \( r^{2} - 15r - 16 \), matching the original trinomial.Employing this method accurately verifies whether the factorization was carried out correctly. It's a reliable check for ensuring all steps lead to the original expression.