Problem 62 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common binomial factor. $$x(y+9)-11(y+9)$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x(y+9) - 11(y+9)\) using greatest common binomial factor is \((y+9)(x - 11)\).

Step 1: Identify the Common Binomial Factor

Look at the terms within the polynomial to identify the greatest common binomial factor. The polynomial can be rewritten to clearly see the common binomial factor; \((y+9)\) is common in both terms i.e., \(x(y+9)\) and \(-11(y+9)\). This gives \(x(y+9) - 11(y+9)\), clearly showing that \((y+9)\) is the common factor.

Step 2: Factor out the Common Binomial

Use the distributive property, which says that \(a(b+c) = ab + ac\), in the reverse order to factor out the common binomial. Take the common binomial factor \((y+9)\) out from the polynomial \(x(y+9) - 11(y+9)\). This gets factored to \((y+9)(x - 11)\).

Step 3: Verifying the Solution

To make sure that the factorization is done correctly, expand the factored polynomial \((y+9)(x - 11)\). When we expand we get back \(x(y+9) - 11(y+9)\), which was the original polynomial.

Key Concepts

Understanding the Greatest Common Factor in Polynomials

The term 'greatest common factor' (GCF) applies not only to numbers but also to expressions such as polynomials. In a mathematical expression, GCF refers to the largest factor that divides each term of the polynomial without any remainder.
To identify the GCF in polynomial expressions, follow these steps:

• Look for common terms or factors in each part of the polynomial. For instance, if in the polynomial expression \(x(y+9) - 11(y+9)\), the factor \((y+9)\) is present in both terms, it becomes the GCF.
• Remember that this factor can be a number, variable, or in this case, a binomial.
• Once identified, you can "factor out" this GCF from the entire polynomial to simplify the expression.

Identifying the GCF helps in the simplification and factorization process, making complex polynomials easier to handle.

Recognizing and Using Common Binomials

A common binomial is essentially a binomial expression that appears in multiple terms of a polynomial. Identifying a common binomial is crucial for simplifying expressions and is a common step in the factorization process.
In our example, \(x(y+9) - 11(y+9)\), the binomial \((y+9)\) occurs in both terms. This can be likened to the concept of the greatest common factor, but extends to expressions or groups of terms.

• Once the common binomial is identified, you can factor it out. This works similarly to factoring out a single term.
• Factoring out the common binomial allows you to restate the polynomial concisely, as shown: \((y+9)(x - 11)\).

By effectively using common binomials, you can transform and simplify expressions, making them far easier to manipulate and understand. This technique is especially useful with more complex polynomials involving multiple terms and factors.

Applying the Distributive Property in Reverse

The distributive property is a fundamental algebraic principle that states \(a(b + c) = ab + ac\). However, in factorization, we often use this property in reverse. Instead of distributing, we factor out the common term.
For the expression \(x(y+9) - 11(y+9)\), you recognize \((y+9)\) as a common factor and factor it out, simplifying the expression to \((y+9)(x - 11)\). This is the reverse application of the distributive property.

• Identify terms that are shared to apply the reverse distributive process effectively.
• Use this property to factor and thus rewrite the polynomial with a common factor grouped outside.
• This technique simplifies expressions, making them easier to work with especially in solving equations.

Understanding how to reverse the distributive property is key in polynomial factorization, as it simplifies expressions by grouping shared factors, leading to simpler and more manageable equations.