Problem 46 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ (5-v) X+(5-v) Y $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial $$r^2 - r - 6$$ is $$(r-3)(r+2)$$.

Step 1: Identify the constant term and the linear term coefficient

The constant term in the given polynomial is -6, and the linear term coefficient is -1.

Step 2: Find two numbers whose product equals the constant term and their sum equals the linear term coefficient

Consider the numbers -3 and 2. Their product is equal to the constant term (-6) and their sum is equal to the linear term coefficient (-1): (-3) * 2 = -6 and (-3) + 2 = -1

Step 3: Write the polynomial as a product of two binomials

Now that we've found the two numbers, we can write the given polynomial as a product of two binomials. Since -3 and 2 are the numbers we're looking for, the factored form will be: $$ (r-3)(r+2) $$ So, the factored form of the polynomial $$r^2 - r - 6$$ is $$(r-3)(r+2)$$.

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients. They are strung together by addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can have one or several terms, each of which consists of a product of a number (the coefficient) and a variable raised to an exponent.

For example, the polynomial expression \( r^2 - r - 6 \) includes three terms: \( r^2 \) which has an implied coefficient of 1, \( -r \) where the coefficient is -1, and the constant term \( -6 \). Understanding the structure of polynomial expressions is a foundational skill for solving and factoring them.

Binomials

A binomial is a polynomial with exactly two terms. Each term is a monomial, and when combined, they create this special subset of polynomials. Binomials are especially important when learning to factor because they often represent factors of more complex polynomials.

When factoring the polynomial \( r^2 - r - 6 \), we aim to break it down into the product of two binomials. The factored form \( (r-3)(r+2) \) shows how two simpler binomials, when multiplied, can produce the original polynomial. Recognizing possible binomial factors is key to the factoring process.

Coefficient Identification

Coefficient identification involves finding the numerical factor that multiplies a variable in a polynomial term. In most cases, the coefficient is directly in front of the variable, but it can also be an implied number if visibly missing.

For instance, in the polynomial \( r^2 - r - 6 \), the coefficient of the \( r^2 \) term is 1 (even though it is not explicitly written), and the coefficient of the \( r \) term is -1. Accurately identifying these coefficients is crucial in the factoring process because they determine the products and sums needed to find the binomial factors.

Constant Term

The constant term in a polynomial is the term that does not contain any variables---it's a standalone number. In the process of factoring polynomials, the constant term plays a significant role since it's the product of the constants of the binomial factors.

In our example, \( r^2 - r - 6 \), the constant term is -6. During factoring, we search for two numbers whose product is the constant term and whose sum is the coefficient of the linear term. Identifying the constant term quickly allows you to focus on finding its factor pairs that meet the required criteria for successful factoring.