Problem 51 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ r\left(z^{2}-6\right)+\left(z^{2}-6\right) $$

Step-by-Step Solution

Answer

The factored expression is \((z^2 - 6)(r + 1)\).

Step 1: Identify Common Terms

Look at the given polynomial expression, which is \( r(z^2 - 6) + (z^2 - 6) \). Notice that both terms have a common factor of \( (z^2 - 6) \). Common terms can be factored out to simplify the expression.

Step 2: Factor Out the GCF

Since both terms contain \( (z^2 - 6) \) as a common factor, factor \( (z^2 - 6) \) from both terms. This gives: \[ (z^2 - 6)(r + 1) \].

Step 3: Verify the Solution

To ensure correctness, distribute \( (z^2 - 6) \) back through \( (r + 1) \). This should yield the original expression: \( r(z^2 - 6) + (z^2 - 6) \). Thus, the factorization is verified.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is an important concept in algebra, especially when dealing with polynomials. The GCF is the largest expression that can divide each term of the polynomial without leaving a remainder.

Understanding how to identify the GCF is crucial as it forms the foundation of simplifying expressions by factoring them. When looking at a polynomial, you search for common factors in all terms. Finding the GCF involves the following steps:

• Examine each term in the polynomial carefully.
• Identify the factor that appears in all terms.
• Choose the highest power of this factor shared by the terms.

In the expression given, \( r(z^2 - 6) + (z^2 - 6) \), your task is to observe that \((z^2 - 6)\) is a repeated factor across both terms. This repetition makes \((z^2 - 6)\) the GCF. By factoring this out, we simplify the original expression into its components, making it easier to handle for further operations.

Polynomial Expressions

Polynomial expressions can seem tricky at first, but breaking them down makes them more approachable. A polynomial is simply an algebraic expression consisting of variables and coefficients, which are combined using addition, subtraction, multiplication, and sometimes division (as long as it's not by a variable).

In the context of our exercise, understanding polynomial structure helps in recognizing patterns, terms, and common factors, which are essential in simplification and factorization.

• Polynomials are identified by the number of terms they have, such as monomials (one term), binomials (two terms), and trinomials (three terms).
• The degree of a polynomial is determined by the highest power of the variable in the expression.
• In the given exercise, each term of the polynomial contains the common binomial \((z^2 - 6)\).

Recognizing the structure of a polynomial can significantly ease the factorization process, as it directs your focus to common factors and terms that can be combined or simplified.

Simplifying Expressions

Simplifying expressions is about making them more manageable and easier to work with. The key aim is to reduce the expression to its simplest form without changing its value. Simplification includes processes like factoring, expanding, and combining like terms.

Factoring, in particular, is pivotal when dealing with polynomials, as finding common factors leads to a simpler representation of the original expression.

• Identify any like terms or common factors that exist in the different parts of the expression.
• Use the GCF to factor out the commonalities, thereby reducing complexity.
• Re-examine what remains to ensure no further simplification can be made.

In our example, simplifying by factoring out the GCF \((z^2 - 6)\) condenses the polynomial from \( r(z^2 - 6) + (z^2 - 6) \) to \((z^2 - 6)(r + 1)\). This makes calculations clearer and sets the stage for further mathematical manipulation if needed. Remember, simplification increases clarity, whether you're solving equations or performing polynomial operations.