Problem 80 - Beginning and Intermediate Algebra

Question

Factor completely. You may need to begin by taking out the GCF first or by rearranging terms. $$10 h k^{3}-5 h k^{2}+30 k^{3}-15 k^{2}$$

Step-by-Step Solution

Answer

The completely factored expression is: \(5k^{2}(h + 3)(2k - 1)\).

Step 1: Identify the GCF of the terms in the expression

List the prime factors of each coefficient (10, 5, 30, 15), and look for the common variable factor of the terms in the expression: \[10 = 2 \times 5\] \[5 = 5\] \[30 = 2 \times 3 \times 5\] \[15 = 3 \times 5\] The GCF of the numerical coefficients is 5, and there's also a common variable factor of \(k^{2}\) in the terms.

Step 2: Factor the GCF out of the expression

We will now factor out the GCF we found in the first step (5 and \(k^{2}\)) from each term in the expression: \[5k^{2}(2hk - 1h + 6k - 3)\]

Step 3: Rearrange the terms within the parentheses

We can reorder the terms so that the like terms are close together, making it easier to spot opportunities for further factoring: \[5k^{2}(2hk + 6k - 1h - 3)\]

Step 4: Factor by grouping the terms inside the parentheses

Now, we will factor by grouping the terms inside the parentheses: \[5k^{2}[(2hk + 6k) + (-1h - 3)]\] We can now factor out a 2k from the first group and -1 from the second group: \[5k^{2}[2k(h + 3) - 1(h + 3)]\]

Step 5: Factor out the common binomial factor

Now that we have a common binomial factor of (h + 3) in both terms inside the parentheses, we can factor that out: \[5k^{2}(h + 3)(2k - 1)\]

Step 6: Write the final factored expression

Now that we have factored the expression completely, we write the final factored expression: \[5k^{2}(h + 3)(2k - 1)\] Thus, the complete factorization of the given expression is: \[5k^{2}(h + 3)(2k - 1)\]

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is the first essential step in polynomial factorization. The GCF of a polynomial refers to the largest factor that can divide each term of the polynomial without leaving a remainder. To identify the GCF of a polynomial, especially one with numeric and variable factors, start by listing the prime factors of each coefficient in the expression.
For example, in the expression \(10 h k^{3} - 5 h k^{2} + 30 k^{3} - 15 k^{2}\), we list them as follows:
\[10 = 2 \times 5\]
\[5 = 5\]
\[30 = 2 \times 3 \times 5\]
\[15 = 3 \times 5\]
This shows that 5 is a factor for each term and can be factored out. Don't forget to also identify any common variable factors, such as \(k^{2}\) here.
Finding and factoring out the GCF simplifies the polynomial, making further factorization much easier.

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four or more terms. This technique is beneficial when identifying common factors in pairs of terms within the polynomial. The idea is to rearrange the terms, if necessary, and then to group them in such a way that each group contains a common factor.
For instance, after factoring out the GCF from our example, we have \(5k^{2}(2hk + 6k - 1h - 3)\). We can reorganize and group as follows: \(5k^{2}[(2hk + 6k) + (-1h - 3)]\).
Within each bracket, we can take advantage of any common factors. In the first group, \(2k\) is common, and in the second group, \(-1\) is common.

• This gives us \(5k^{2}[2k(h + 3) - 1(h + 3)]\).

Remember that successful grouping results in a common binomial factor in both groups, which can be factored out completely.

Prime Factors

Understanding prime factors is key to many factoring processes, including finding a GCF. A number's prime factors are the set of prime numbers that multiply together to give the original number. This is vital in identifying the GCF of numeric coefficients within a polynomial.
For example, if you look at the numbers 10, 5, 30, and 15 in our polynomial expression, breaking them down to their prime factors, you get:

• \(10 = 2 \times 5\)
• \(5 = 5\)
• \(30 = 2 \times 3 \times 5\)
• \(15 = 3 \times 5\)

Identifying the common prime factors, in this case, \(5\), allows you to know the numeric part of the GCF. This helps in effectively simplifying polynomial expressions, making them easier to factor completely.

Variable Factors

When factoring polynomials, it's important to consider not only the numeric factors but also the variable factors. This involves identifying variables raised to powers that are common across terms.
In the polynomial \(10 h k^{3} - 5 h k^{2} + 30 k^{3} - 15 k^{2}\), the variable \(k\) appears in every term. Observing the lowest power of \(k\), in this case, \(k^{2}\), allows us to factor out \(k^{2}\) as part of the GCF.
Here's how variable factors work:

• Look at each term and identify the lowest power of common variables.
• Factor these out along with the numerical GCF.

By exploring both the numeric and variable components, you can simplify the expression and make further steps like grouping much easier. The careful identification of variable factors ensures a complete and thorough factorization process.