Problem 6 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 6 a c+3 a b+6 b c+3 b^{2} $$

Step-by-Step Solution

Answer

3(a + b)(2c + b)

Step 1: Group the terms

Group the terms in pairs to factor by grouping: (6ac + 3ab) + (6bc + 3b^2)

Step 2: Factor out the greatest common factor (GCF) from each pair

Factor out the GCF from each pair: 3a(2c + b) + 3b(2c + b)

Step 3: Factor out the common binomial factor

Since both terms contain the factor (2c + b), factor it out: (2c + b)(3a + 3b)

Step 4: Simplify

Notice that 3a + 3b can be factored further: (3)(a + b). Thus, the expression will be: (2c + b)(3)(a + b) Therefore, the completely factored form is: 3(a + b)(2c + b).

Key Concepts

greatest common factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. To find the GCF for each group, we look for the highest number and variable(s) that can evenly divide each term.
For example:

• In the expression 6ac + 3ab, 3a is the GCF because 3 is the highest number that divides 6 and 3, and 'a' is common to both terms.
• Similarly, for 6bc + 3b^2, the GCF is 3b because both terms can be divided by 3 and have at least one b.

When factoring, we often group terms to identify the GCF in smaller chunks. For instance, starting with the original expression 6ac + 3ab + 6bc + 3b^2, we grouped as follows: (6ac + 3ab) + (6bc + 3b^2). Then we factored out the GCF from each group: 3a(2c + b) + 3b(2c + b).
Identifying and factoring out the GCF is a crucial first step to simplifying polynomials.

factoring by grouping

Factoring by grouping involves splitting a polynomial into smaller groups, factoring out the GCF from each group, and then factoring out any common binomial factors. This technique is useful when a polynomial has four or more terms.
For the expression 6ac + 3ab + 6bc + 3b^2:
Step 1: Group terms into pairs: (6ac + 3ab) + (6bc + 3b^2).
Step 2: Factor out the GCF from each group:

• The GCF of 6ac + 3ab is 3a: 3a(2c + b)
• The GCF of 6bc + 3b^2 is 3b: 3b(2c + b)

Step 3: Factor out the common binomial factor (2c + b):
(2c + b)(3a + 3b)
Step 4: Simplify further if possible. In this case, 3a + 3b can be factored further into 3(a + b):
(2c + b)(3)(a + b)
Factoring by grouping breaks down complex polynomials into more manageable pieces and is especially helpful when no single GCF can be factored out from the entire polynomial.

prime polynomials

A prime polynomial is one that cannot be factored into the product of other polynomials with integer coefficients (other than 1 and the polynomial itself).
For instance, if we cannot simplify a polynomial into smaller factors, it is considered prime.
In our example, after completely factoring 6ac + 3ab + 6bc + 3b^2, we see it simplifies to 3(a + b)(2c + b). Each factor cannot be broken down further (without using fractions or radicals), indicating that none of the components is a prime polynomial.
Prime polynomials are the building blocks of other polynomials, similar to how prime numbers are fundamental in number theory.
Recognizing prime polynomials is crucial because it signals the end of the factoring process. If a polynomial can't be factored further, it is presented in its simplest form.