Problem 76 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out a negative common factor. $$-r^{2}+11 r-28$$

Step-by-Step Solution

Answer

The short answer to factoring the expression completely by first taking out a negative common factor is: \(-1(r - 4)(r - 7)\).

Step 1: Identify negative common factors

In the given expression \(-r^2 + 11r - 28\), the negative common factor is -1. So, we first need to multiply the entire expression by -1.

Step 2: Multiply the expression by -1

Multiplying the expression by -1, we get: \((-1) * (-r^2 + 11r - 28)\) \(r^2 - 11r + 28\)

Step 3: Factor the quadratic expression

Now, we need to factor the quadratic expression \(r^2 - 11r + 28\). We are looking for two numbers that multiply to 28 and, when added, give us -11. These numbers are -4 and -7. So, we can rewrite the expression as: \((r - 4)(r - 7)\)

Step 4: Write down the final factored expression

The original expression, \(-r^2 + 11r - 28\), can now be written in factored form as: \(-1(r - 4)(r - 7)\) The final factored expression is \(-1(r - 4)(r - 7)\).

Key Concepts

Negative Common Factor

Finding a negative common factor is often the first step in factoring certain polynomial expressions, especially when you start with a negative leading term. The negative leading term can make factoring more complex, so isolating that negative can simplify the process. In our example, we looked at the expression

• -r² + 11r - 28

by identifying -1 as a negative common factor. By factoring out -1, we change how the expression looks, making it easier to manage:

• -1(r² - 11r + 28)

This does not change the value of the expression, but it makes the math more manageable by eliminating the negative sign from the leading coefficient. This step is crucial since it sets up the expression for easier factorization in the subsequent steps.

Quadratic Expression

Quadratic expressions are polynomials of the form \(ax^2 + bx + c\). These are equations where the highest degree of the variable is 2. They have significant meaning in algebra because they represent parabolas when plotted on a graph. In our exercise, we're dealing with

• r² - 11r + 28

which is obtained after factoring out the negative common factor.

The primary goal is to break down this quadratic expression into products of simpler binomials. This means we're looking for two binomials that multiply to form the original quadratic.

Identifying the right pair of numbers involves checking factors of the constant term (28, in this case) that also add up to the linear coefficient (-11). The solution was determined as (-4 and -7), and this gave:

• (r - 4)(r - 7)

Polynomial Factoring

Factoring polynomials involves finding simpler expressions that multiply together to recreate the original polynomial. This process is crucial in solving polynomial equations, simplifying expressions, and performing algebraic operations. The expression

• -r² + 11r - 28

from our exercise was split into simpler parts using the following two key steps:

• First, extracting the negative common factor resulted in: -1 (r² - 11r + 28).
• Second, factoring the quadratic \(r² - 11r + 28\) into: (r - 4)(r - 7).

Together, these steps give us the complete factored form of the original polynomial:

• -1(r - 4)(r - 7)

This method transforms a more complex polynomial into the product of simpler, easier-to-manage expressions. Recognizing the structure of polynomials and the relationship between their terms is key to successful factoring.