Problem 60 - Elementary Algebra

Question

For the following problems, factor the polynomials, if possible. $$ 27 r^{2}-33 r-4 $$

Step-by-Step Solution

Answer

Answer: \((9r - 1)(3r + 4)\)

Step 1: Check for Common Factors

First, let's check if there are any common factors for all the terms in the given polynomial: \(27r^2 - 33r - 4\). We can see that there are no common factors among the coefficients, so we can proceed to the next step.

Step 2: Factoring by Grouping

Since we couldn't find any common factors in the previous step, let's try factoring by grouping. To do this, we will need to find two numbers whose product is equal to the product of the first and last coefficients (i.e., \(27 \times (-4) = -108\)) and whose sum is equal to the middle coefficient (i.e., \(-33\)) Let's list the factors of -108: 1. (1, -108) 2. (2, -54) 3. (3, -36) 4. (-1, 108) 5. (-2, 54) 6. (-3, 36) The only pair of factors that satisfies the condition of having a sum of -33 is (-3, 36). Hence, we rewrite the polynomial as follows: $$ 27r^2 - 3r + 36r - 4 $$ Now, we can try to group the terms and factor out any common factors: $$ 3r(9r - 1) + 4(9r - 1) $$ Now, we see that both terms have a common factor of \((9r - 1)\). So we can factor it out:

Step 3: Final Factored Form

Now we can express the factored form of the polynomial: $$ (9r - 1)(3r + 4) $$ So the factored form of the given polynomial \(27r^2 - 33r - 4\) is \((9r - 1)(3r + 4)\).

Key Concepts

Common Factors

Understanding common factors is a fundamental aspect of factoring polynomials. A common factor refers to a number or variable that divides each term of the polynomial without leaving a remainder. In simpler terms, it's a shared ingredient in all parts of the equation.

For example, in the expression \(2x^2 + 4x + 6\), the common factor is 2, because 2 can be evenly divided out of each term. Finding common factors is the first step in simplifying polynomials. However, if a polynomial like \(27r^2 - 33r - 4\) shows no apparent common factor as seen in our exercise, we need to proceed with other factoring techniques.

Factoring By Grouping

When factoring by grouping, we look for ways to reorganize and group terms in a polynomial to reveal common factors. Typically, this method works well with polynomials that have four or more terms. The primary goal is to create groups that each have a common factor we can extract.

In our exercise example, we started with the trinomial \(27r^2 - 33r - 4\) and found two numbers that product to \(27 \times (-4)\) and add to -33, which are -3 and 36. We then rewrote the trinomial as two binomials: \(27r^2 - 3r\) and \(36r - 4\). By doing this, we effectively grouped the polynomial into parts that each contain a common factor, making it easier to factor further.

Polynomial Coefficients

The numbers that multiply the variables in a polynomial are called polynomial coefficients. These numbers are critical when factoring because they determine the potential factors of a polynomial and their relationships. In the case of the given problem, our coefficients are 27, -33, and -4 for \(27r^2\), \(33r\), and \(4\), respectively.

The relationship between these coefficients plays a significant role in determining the strategy to factor the polynomial. For the problem at hand, the product of the coefficients of the first and last terms, \(27 \times -4 = -108\), and the middle coefficient, -33, guided the process of finding suitable numbers for factoring by grouping.

Writing Factored Form

The transformation of a polynomial into a product of simpler polynomials or factors is what we call the factored form. Writing polynomials in factored form makes them easier to work with, especially when solving equations.

For our polynomial \(27r^2 - 33r - 4\), after successfully utilizing factoring by grouping, we identified a common binomial factor, \(9r - 1\), from the grouped terms. Once we've grouped the terms and factored out the common binomial, the polynomials take on their factored form, exemplified in the expression \(9r - 1)(3r + 4)\). This not only simplifies the polynomial but also prepares it for further mathematical operations, such as finding roots or simplifying rational expressions.