Problem 83 - Beginning and Intermediate Algebra

Question

Factor completely. You may need to begin by taking out the GCF first or by rearranging terms. $$42 k^{3}+15 d^{2}-18 k^{2} d-35 k d$$

Step-by-Step Solution

Answer

The completely factored expression is \((7k - 3d)(6k^2 - 5d)\).

Step 1: Rearrange the terms

Rearrange the terms so that they are in decreasing order of powers of the variables: \(42k^3 - 18k^2d + 15d^2 - 35kd\)

Step 2: Identify the GCF of each term

We can identify the GCF of each term in the expression: - For the first two terms, the GCF is \(6k^2\). - For the last two terms, the GCF is \(5d\).

Step 3: Factor out the GCF for each pair of terms

Factor out the GCF of each pair of terms: \(6k^2(7k - 3d) + 5d(3d - 7k)\)

Step 4: Rearrange terms in the second parenthesis

Rearrange the terms in the second parenthesis so that it has the same order as the first parenthesis: \(6k^2(7k - 3d) + 5d(-7k + 3d)\)

Step 5: Factor out a negative sign in the second parenthesis

Factor out a negative sign in the second parenthesis to match the first parenthesis: \(6k^2(7k - 3d) - 5d(7k - 3d)\)

Step 6: Factor the common binomial

Now, we can factor the common binomial (7k - 3d) from the expression: \((7k - 3d)(6k^2 - 5d)\) The completely factored expression is \((7k - 3d)(6k^2 - 5d)\).

Key Concepts

GCF (Greatest Common Factor)

The Greatest Common Factor, often abbreviated as GCF, is a critical concept in simplifying and factoring polynomials. Think of it as the largest factor that evenly divides two or more numbers or terms. When factoring expressions, identifying the GCF is a crucial first step because it can help simplify the expression into simpler terms that are easier to work with.

To find the GCF of a set of terms, you should consider the numerical factors and the algebraic variable factors separately.

• For the numerical part, identify the highest number that can divide all the coefficients of the terms.
• For the variables, use the lowest power of any common variable present in all terms.

Let's consider an example:

• If you have terms like \(42k^3\) and \(-18k^2d\), break it down by taking the GCF of 42 and 18, which is 6. For the variable part, \(k\) is common, and the lowest power is \(k^2\).

Given this, the GCF for \(42k^3\) and \(-18k^2d\) is \(6k^2\). Similarly, handle the next set of terms \(15d^2\) and \(-35kd\) by extracting \(5d\) as their GCF. By spotting these GCFs and factoring them out, you play a key role in simplifying the task of polynomial factoring.

binomial factoring

Binomial factoring is a method used to factor expressions that contain two terms, usually in a form where they can be grouped to show a common factor. In polynomial expressions, once the initial GCF is factored out from the terms, they oftentimes reveal binomials which can be further factored.

The goal is to identify a common binomial factor in what remains of the factored expression. Once identified, this common binomial expression can be factored out of the equation, simplifying it further.

• For instance, consider the expression \(6k^2(7k - 3d) + 5d(3d - 7k)\).
• By carefully examining, we see that both groups encompass the pattern \((7k - 3d)\); however, the second is disguised with reversed signs.
• Factoring out a negative from \(5d(3d - 7k)\) gives us \(5d(-7k + 3d)\), which then matches the structure of \(6k^2(7k - 3d)\).

After aligning both parenthesis terms, the binomial \((7k - 3d)\) can be factored out. Following this, the polynomial simplifies to involve one binomial and another trinomial or polynomial expression, which is the essence of binomial factoring.

rearranging polynomials

Rearranging polynomials is a strategic approach used to sequence terms within an expression to reveal opportunities for factoring or simplification. The order that you arrange terms in can greatly influence the ease with which you can factor an expression. Typically, you arrange terms based on the powers of variables or according to how they can conveniently reveal a common factor.

In our original exercise, rearranging was carried out by positioning terms so that those with higher powers, such as \(k^3\), were placed at the start. Once rearranged - often in descending order - the terms were more clearly viewed in pairs that hinted at possible factors.

• This involved regrouping the polynomial terms from \(42 k^3+15 d^2-18 k^2 d-35 k d\)
• to \(42k^3 - 18k^2d + 15d^2 - 35kd\).

This rearranged format makes it more straightforward to group terms and determine the GCF, thus facilitating a streamlined path to factoring. Hence, rearranging plays an instrumental role in simplifying complex polynomial expressions by preparing them for subsequent steps of the factoring process.