Problem 47 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 18 a^{2}-6 a b+42 a c-14 b c $$

Step-by-Step Solution

Answer

The factored expression is \(2(3a - b)(3a + 7c)\).

Step 1: Identify Common Factors

Look at the expression: \(18a^2 - 6ab + 42ac - 14bc\). Identify any common factors in all terms. Here, each term can be divided by 2, which is a common factor. Factor out 2 from each term to get: \(2(9a^2 - 3ab + 21ac - 7bc)\).

Step 2: Group Terms

Now, with the simplified expression \(9a^2 - 3ab + 21ac - 7bc\), group terms to see if further factoring is possible. Try grouping as: \((9a^2 - 3ab)\) and \((21ac - 7bc)\).

Step 3: Factor by Grouping First Pair

Consider the first group \((9a^2 - 3ab)\). Factor out the greatest common factor, which is 3a, giving: \(3a(3a - b)\).

Step 4: Factor by Grouping Second Pair

Now, for the second group \((21ac - 7bc)\), factor out the greatest common factor, which is 7c, resulting in: \(7c(3a - b)\).

Step 5: Combine the Two Results

Notice that \((3a - b)\) is a common binomial factor in both groups. The expression can now be factored as: \( (3a - b)(3a + 7c)\).

Step 6: Write Final Factorization

Combine the common factor of 2 from Step 1 with the binomial factors obtained: \(2(3a - b)(3a + 7c)\). This is the fully factored form of the original expression.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept used in factoring algebraic expressions. It refers to the largest number or expression that can evenly divide all terms in a given polynomial. By identifying the GCF, you simplify the process of factoring expressions and set the stage for more complex operations like factoring by grouping.

Consider the expression:

• \(18a^2 - 6ab + 42ac - 14bc\)

Start by examining each term to find a number and any variable factor they all share. Here:

• \(18, -6, 42,\) and \(-14\) share the number 2 as a factor. So, 2 is the GCF of these coefficients.

Next, divide each term by 2:

• \(18a^2 \rightarrow 9a^2\)
• \(-6ab \rightarrow -3ab\)
• \(42ac \rightarrow 21ac\)
• \(-14bc \rightarrow -7bc\)

This gives us a new expression: \(2(9a^2 - 3ab + 21ac - 7bc)\). This initial simplification is crucial for the subsequent factoring processes.

Factoring by Grouping

Factoring by grouping exploits the structure of an expression by rearranging terms into smaller groups, which can then be factored individually. This method is especially useful when there isn't an obvious common factor across all terms in a polynomial.

For the expression \(9a^2 - 3ab + 21ac - 7bc\), we can group terms to reveal pairs that have their own common factors:

• Group 1: \(9a^2 - 3ab\)
• Group 2: \(21ac - 7bc\)

For Group 1,

• Factor out the GCF, which is 3a: \[3a(3a - b)\]

For Group 2,

• Factor out the GCF, which is 7c: \[7c(3a - b)\]

Notice that \((3a - b)\) appears in both groups, indicating a common factor across these factored groups, which enriches the factoring-by-grouping method and allows for further simplification.

Algebraic Simplification

Algebraic simplification wraps up the factoring process by combining intermediate results into a final, neat expression. This step underscores the power of simplification in revealing the inherent structure of algebraic expressions.

After using the method of factoring by grouping, you find that both grouped terms share the common binomial factor

• \((3a - b)\).

You then create a new expression from these shared elements:

• \((3a - b)(3a + 7c)\)

Finally, do not forget initial factors you found at the beginning, like the GCF of the entire original expression. Here, that was 2. Thus, the complete factored form becomes:

• \(2(3a - b)(3a + 7c)\)

Through algebraic simplification, you refine the expression into its simplest form while maintaining mathematical integrity.