Problem 56 - Algebra A Combined Function

Question

Simplify each expression. Each exercise contains a four-term polynomial that should be factored by grouping. $$ \frac{a b+a c+b^{2}+b c}{b+c} $$

Step-by-Step Solution

Answer

The simplified expression is \(a + b\).

Step 1: Identify Groups

Group the terms of the polynomial to form two pairs. Here, the expression can be grouped as: \((ab + ac) + (b^2 + bc)\). This setup allows us to factor by grouping.

Step 2: Factor Each Group

Factor the common factor from each group. In the first group \((ab + ac)\), the common factor is \(a\), resulting in \(a(b + c)\). In the second group \((b^2 + bc)\), the common factor is \(b\), resulting in \(b(b + c)\). This gives us: \(a(b + c) + b(b + c)\).

Step 3: Factor Out the Common Binomial

The expression now is \(a(b + c) + b(b + c)\), where \(b+c\) is a common binomial factor. Factor \((b+c)\) out, resulting in \((b + c)(a + b)\).

Step 4: Simplify the Fraction

Substitute the factored form back into the original expression: \(\frac{(b + c)(a + b)}{b + c}\). Since \(b+c\) is present in both the numerator and denominator, it can be canceled out, leaving \(a + b\) as the simplified expression.

Key Concepts

Factoring by Grouping

Factoring by grouping is a valuable technique used to simplify polynomials, especially those that contain four terms. This approach involves creating smaller groups from the polynomial to make it easier to find common factors within them. In the context of the original exercise, this technique helped to simplify the expression \( ab + ac + b^2 + bc \).
To begin factoring by grouping, you should first identify and group terms in a way that makes common factors obvious:

• You can break the original polynomial into pairs:
  • First group: \( (ab + ac) \)
  • Second group: \( (b^2 + bc) \)

Next, factor each group individually by identifying and extracting the common factor. Factoring by grouping essentially transforms a complex polynomial into a series of simpler multiplication problems, which can then be easily managed and solved. This method makes the process of finding factors more systematic and less daunting.

Common Factor

Finding the common factor is crucial when simplifying expressions through factoring. A common factor is a factor that is shared by all the terms in a group. In the example given, each group of terms can be simplified by pulling out this shared factor.
For instance:

• In the first group \( (ab + ac) \), the common factor is \( a \). This means we can factor \( a \) out, leaving us with \( a(b + c) \).
• Similarly, in the second group \( (b^2 + bc) \), the common factor is \( b \). So, you can factor \( b \) out, resulting in \( b(b + c) \).

By identifying these common factors, you simplify the expression to a point where further factoring can occur, in this case reducing it to common binomial factors. This step is foundational in the process of polynomial simplification.

Binomial Factorization

Binomial factorization is the process of rewriting a polynomial as a product of two binomial expressions. It often follows factoring by grouping and identifying common factors. In our example, after factoring the groups, the expression became \( a(b + c) + b(b + c) \).
With binomial factors identified, see if there is a common binomial present across all terms:

• In this case, \( (b + c) \) appears in both terms, making it a common factor across them.

Once identified, factor this common binomial out of the expression. The factored expression from our example becomes \( (b + c)(a + b) \).
This step is crucial in simplifying the original polynomial because, when substituted back into the equation, it turns into a simple fraction that can be further simplified by canceling out recurring terms, leading us to the final simplified term: \( a + b \).