Problem 29 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ x^{3}+5 x^{2} $$

Step-by-Step Solution

Answer

The GCF factored form is \(x^{2}(x + 5)\).

Step 1: Identify the Common Term

Look for the greatest common factor (GCF) in the polynomial terms. Here, we have the terms \(x^{3}\) and \(5x^{2}\). The common factor is \(x^{2}\), as it is the highest power of \(x\) common to both terms.

Step 2: Divide each Term by the GCF

Now, divide each term of the polynomial by the identified GCF, \(x^{2}\). - For \(x^{3}\), dividing by \(x^{2}\) gives us \(x\).- For \(5x^{2}\), dividing by \(x^{2}\) gives us \(5\).

Step 3: Write the Factored Expression

Finally, express the polynomial as a product of the GCF and the resulting terms from the division. This gives us: \[ x^{2}(x + 5) \]

Key Concepts

Greatest Common Factor

The Greatest Common Factor, often abbreviated as GCF, is a key concept in factoring polynomials. When dealing with polynomials, the GCF is the largest factor that divides each term of the polynomial without leaving a remainder. Finding this factor is crucial because it simplifies complicated expressions by pulling out a common element, making operations easier. To identify the GCF in polynomial terms, compare each term, and find the highest power of common variables and any common numerical factors.

• For instance, consider the polynomial terms: \(x^3\) and \(5x^2\).
• Both terms have a factor of \(x^2\), and this factor is the highest power common to both.

By factoring out \(x^2\), we simplify the polynomial, making further computations and algebraic manipulations more manageable. This step is the building block for more advanced algebraic processes.

Polynomial Terms

Polynomial terms are the parts of a polynomial expression, separated by addition (+) or subtraction (-) signs. A polynomial is a mathematical expression that includes variables raised to non-negative integer powers, and these expressions can be as simple as a single monomial or as complex as multinomial expressions with multiple variables or terms.Each term in a polynomial consists of:

• A coefficient (which is the numerical part).
• A variable raised to a power (which is the exponent).

For example, in the polynomial \(x^3 + 5x^2\), we have:

• \(x^3\) as a term (with a coefficient of 1 and an exponent of 3).
• \(5x^2\) as another term (with a coefficient of 5 and an exponent of 2).

Understanding each term's components helps in identifying common factors, which is integral to processes like factoring, evaluating, and simplifying polynomial expressions.

Algebraic Expressions

Algebraic expressions encompass a wide range of mathematical phrases that can include numbers, variables, and operations (like addition, subtraction, multiplication, etc.). A polynomial is a type of algebraic expression and serves as a building block of higher mathematical functions and equations. The beauty of algebra lies in its ability to abstractly represent real-world phenomena using these expressions. Factors, terms, and expressions must be maneuvered through various algebraic rules and operations to solve equations, simplify expressions, or perform functions like factoring. When working with algebraic expressions, it's important to:

• Identify like terms, which allows for simplification.
• Utilize the properties of arithmetic operations to combine, decompose, or manipulate terms efficiently.
• Apply distributive, associative, and commutative properties where needed to re-arrange and simplify the expressions effectively.

These skills are not just foundational to understanding algebra, but they also build a bridge to tackle more complex math challenges seamlessly.