Problem 6 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{28 k^{4}+8 k^{3}-40 k^{2}}{4 k^{2}}$$

Step-by-Step Solution

Answer

The simplified expression after dividing the given polynomial is \(7k^2 + 2k - 10\).

Step 1: Factorize the Numerator

In the numerator, we can see that it is a polynomial in terms of k. We will start by factoring out the common factors from each term, which in this case is 4k^2: \[ 28 k^{4}+8 k^{3}-40 k^{2} = 4k^{2}(7k^2 + 2k - 10) \]

Step 2: Factorize the simplified expression in the parentheses

Now that we have factored out our common factor 4k^2, we need to determine if the remaining expression inside the parentheses can be further factorized: \[ 7k^2 + 2k - 10 \] Since no common factor exists for each term and it is not a quadratic that can be factored easily, this expression is considered fully factorized.

Step 3: Simplify the overall expression and cancel out common factors

Now we can rewrite the given expression and factorize the denominator as well, finally cancelling out the common factors: \[ \frac{28 k^{4}+8 k^{3}-40 k^{2}}{4 k^{2}} = \frac{4k^{2}(7k^2 + 2k - 10)}{4 k^{2}} \] Now, we will cancel the common factor of 4k^2: \[ \frac{4k^{2}(7k^2 + 2k - 10)}{4 k^{2}} = 7k^2 + 2k - 10 \]

Step 4: Write down the final answer

The simplified expression after dividing the given polynomial is: \[ 7k^2 + 2k - 10 \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial technique used in algebra to simplify expressions and solve equations. The goal here is to find an equivalent expression that is easier to work with.

In the exercise provided, the numerator is a polynomial:

• 28k^4 + 8k^3 - 40k^2

The first step is to identify any common factors among the terms. A common factor is a variable or number that divides evenly into each term.

Here, each term contains k raised to the power of 2, and each numerical coefficient is divisible by 4. Thus, the greatest common factor (GCF) is 4k².

• This results in factoring the polynomial as: 4k²(7k² + 2k - 10).

Finding the GCF and dividing each term by it simplifies the polynomial for further manipulation.

Simplifying Expressions

Simplifying expressions means rewriting them in their most concise form without changing their value. Once the GCF has been factored out, attention is turned to simplifying what remains.

After factoring out 4k² from the polynomial in the numerator as shown in the exercise, you are left with a simpler expression inside the parentheses:

• 7k² + 2k - 10

Attempt to factor this expression further.

However, checking for further factorization reveals no simple common factors or recognizable patterns like perfect squares or simple trinomials. Therefore, this expression is considered fully simplified and cannot be factored further with ease. That means this expression remains as it is in its simplest form after removal of the GCF.

Canceling Common Factors

Canceling common factors is the process where you reduce a fraction by dividing both the numerator and the denominator by their common factors.

After factoring the numerator, the exercise rewrites the entire fraction:

• \(\frac{4k^{2}(7k^2 + 2k - 10)}{4 k^{2}}\)

Here, 4k² is present in both the numerator and the denominator. Dividing them out simplifies the fraction. The process is akin to multiplying by 1, which retains the fraction’s value while simplifying its expression.

• This leaves the simpler polynomial: 7k² + 2k - 10.

Canceling common factors is handy because it often greatly reduces the complexity of a problem by removing redundant parts of expressions.