Problem 56 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 3 a d+b c+a c+3 b d $$

Step-by-Step Solution

Answer

The factored form of the expression is (a + b)(3d + c).Both (a + b) and (3d + c) are prime polynomials.

Step 1: Identify Groups

Rearrange the terms and group them for easier factoring: 3ad + 3bd + ac + bc

Step 2: Factor Each Group

Factor out the greatest common factor (GCF) from each group. For the first group (3ad + 3bd), the GCF is 3d. For the second group (ac + bc), the GCF is c. This gives us: 3d(a + b) + c(a + b)

Step 3: Factor Out the Common Binomial

Notice that (a + b) is a common factor in both terms. Factor (a + b) out: (a + b)(3d + c)

Step 4: Checking the Factoring

Expand the factored form to ensure it is equal to the original expression. (a + b)(3d + c) = a(3d + c) + b(3d + c) = 3ad + ac + 3bd + bc This matches the original expression, confirming our factoring is correct.

Step 5: Prime Polynomial Identification

Neither (a + b) nor (3d + c) can be factored further; hence they are prime polynomials.

Key Concepts

Prime Polynomials

A polynomial is considered 'prime' if it cannot be factored further over the set of integers. For example, in our exercise, we found the expression \(3ad + bc + ac + 3bd\) steps towards its factorization and ended up with two binomials: \((a + b)(3d + c)\). After examining both binomials, we notice that neither can be broken down into more basic factors. Thus, \(a + b\) and \(3d + c\) are prime polynomials.
This means, when you can't factor a polynomial any further, it is classified as prime. In simpler terms, it's like a number in its simplest form, like 7. You can't divide 7 by anything other than 1 and itself without getting a fraction. For polynomials, once you can't break them down any more, they are termed as 'prime'.

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest expression that can be factored out from two or more terms. Think of it as finding the biggest piece of a puzzle that fits into all parts.
For instance, in our problem, while breaking down \(3ad + 3bd + ac + bc\), the groups \(3ad + 3bd\) and \(ac + bc\) have common factors. We identify the GCF of \(3ad + 3bd\) as \(3d\) and \(ac + bc\) as \(c\). This gives us:
\(3d(a + b) + c(a + b)\).
Identifying the GCF helps simplify expressions and makes it easier to factor polynomials. When you pull out the GCF, you're essentially taking a common divisor that simplifies the entire structure of the polynomial.
This step is essential in polynomial factorization because it reduces the polynomial into a more easily handled form.

Factoring Techniques

Factoring techniques are systematic methods applied to break down expressions into simpler multiplicative forms. One common approach is 'factoring by grouping' used in our current problem. Let's look at this process in detail:

• Rearrange and Group: This means moving terms around and grouping them to make it easier to factor. Our expression \(3ad + bc + ac + 3bd\) can be rearranged to \(3ad + 3bd + ac + bc\).
  
• Factor Each Group: Look for the GCF in each group. Here, we factor \(3d\) out of \(3ad + 3bd\), and \(c\) out of \(ac + bc\). This turns our expression into: \(3d(a + b) + c(a + b)\).
  
• Factor Out the Common Binomial: Notice that \(a + b\) is a common factor in both terms. Factoring \(a + b\) out, we get: \((a + b)(3d + c)\).
  

These steps reveal the underlying structure and help further simplification. Factoring techniques simplify complex expressions, making it easier to solve polynomial equations.

Binomial Factoring

Binomial factoring is a technique for breaking down polynomials into products of binomials (expressions with two terms). Our exercise revolves around this concept, where the final factored form of \(3ad + bc + ac + 3bd\) is \((a + b)(3d + c)\). Here’s a simple breakdown of binomial factoring:

• Identify Common Factors: Look for terms in the polynomial that frequently appear together. For example, \(a + b\) appears in both grouped terms.
  
• Factor Out the Binomial Common Factor: Once identified, factor the binomial out of the entire polynomial. This gives a product of binomials.
  

Binomial factoring simplifies complex expressions, enabling easier manipulation and solving of polynomial equations.
Understanding binomial factoring helps grasp polynomial identities, which are foundational for algebra and calculus.