Problem 62 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 8 c z+10 d z+12 a c+15 a d $$

Step-by-Step Solution

Answer

The polynomial factorizes to \((2z + 3a)(4c + 5d)\). The factorization is correct upon verification.

Step 1 - Group the terms

Group the terms in pairs to make factoring easier: \( (8cz + 10dz) + (12ac + 15ad) \).

Step 2 - Factor out the greatest common factor (GCF) from each group

For the first group \(8cz + 10dz\), the GCF is \(2z\). For the second group \(12ac + 15ad\), the GCF is \(3a\). Factoring out these GCFs gives: \(2z(4c + 5d) + 3a(4c + 5d)\).

Step 3 - Factor out the common binomial

Notice that \(4c + 5d\) is a common factor in both terms. Factor \(4c + 5d\) out: \((2z + 3a)(4c + 5d)\).

Step 4 - Check the factorization

Expand \((2z + 3a)(4c + 5d)\) to check if it matches the original polynomial: \((2z + 3a)(4c + 5d) = 2z(4c + 5d) + 3a(4c + 5d) = 8cz + 10dz + 12ac + 15ad\). The original polynomial is correctly factorized.

Key Concepts

Grouping Method

When factoring polynomials, one effective technique is the grouping method. This is especially useful if you cannot immediately spot a common factor for all terms. Here's how it works: First, look at the polynomial and split it into groups of terms. In our example, we have four terms: \(8cz + 10dz + 12ac + 15ad\). By grouping similar terms together, you get:

• (8cz + 10dz) - These terms have a 'z' in common.
• (12ac + 15ad) - These terms involve 'a' and 'd'.

Grouping makes it easier to apply further factoring techniques and find common factors in smaller segments of the polynomial.

Greatest Common Factor

Once you have grouped your terms, the next step is to factor out the greatest common factor (GCF) from each group. The GCF is the largest factor that divides each term. Consider the first group in our example:

• Group: (8cz + 10dz)
• The GCF is: 2z

Factoring this out, you get \(2z(4c + 5d)\). Now, let's look at the second group:

• Group: (12ac + 15ad)
• The GCF is: 3a

Factor this out to get \(3a(4c + 5d)\). By factoring out the GCF, you've simplified each part of the polynomial, making it easier to identify common factors across the entire expression.

Factoring Binomials

After factoring out the greatest common factors, you often end up with similar terms which can form a common binomial factor. In our example, after factoring out the GCFs, we have: \(2z(4c + 5d) + 3a(4c + 5d)\). Notice that both terms now contain the binomial \(4c + 5d\). You can factor this common binomial out, treating it as a common factor:

• Binomial: (2z + 3a)
• Common binomial factor: (4c + 5d)

Factoring the common binomial out results in: \((2z + 3a)(4c + 5d)\). This is the fully factored form of the original polynomial.

Checking Factorization

To ensure the factorization is correct, always check by expanding the factors and confirming that you get the original polynomial. For our example, we factored the polynomial into \((2z + 3a)(4c + 5d)\). Expand this product:

• First, apply the distributive property: \(2z \cdot 4c + 2z \cdot 5d + 3a \cdot 4c + 3a \cdot 5d\)
• Simplify: \(8cz + 10dz + 12ac + 15ad\)

Since the expanded form matches the original polynomial \(8cz + 10dz + 12ac + 15ad\), the factorization is correct. Always performing this check helps you ensure accuracy and understand the relationships between the factors.