Problem 87 - Elementary Algebra

Question

Either factor out the greatest common factor or factor by grouping. $$ a^{3}+a^{4}+a^{5}+a^{7} $$

Step-by-Step Solution

Answer

a^{3}(1 + a + a^{2} + a^{4})

Step 1: Identify the Terms

Identify all the terms in the given polynomial: a^{3}, a^{4}, a^{5}, a^{7}

Step 2: Find the Greatest Common Factor

Examine each term to find the greatest common factor (GCF). The GCF of all terms is the smallest power of a that appears in each term, which in this case is a^{3}.

Step 3: Factor Out the GCF

Factor out the GCF ( a^{3}) from each term in the polynomial:a^{3}(1 + a + a^{2} + a^{4})

Step 4: Simplify the Expression

The simplified expression after factoring out the GCF is: a^{3}(1 + a + a^{2} + a^{4})

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an important concept when factoring polynomials. It is the largest factor that divides all terms in the polynomial uniformly. To find the GCF, look at each term and determine the lowest power of any variable present in all of them. In the exercise given, the terms were:

• \(a^{3}\)
• \(a^{4}\)
• \(a^{5}\)
• \(a^{7}\)

The smallest power of \(a\) here is \(a^{3}\). Hence, this power is the GCF for these terms. Identifying the GCF is the first step in simplifying and solving many algebraic expressions, as it 'unlocks' the common factor, making the rest of the polynomial easier to handle.

Factoring Techniques

Factoring techniques are methods used to rewrite a polynomial as a product of simpler expressions. The problem provided was solved using a basic but essential factoring technique: factoring out the GCF. Here's a breakdown: First, recognize each term: \(a^{3}\), \(a^{4}\), \(a^{5}\), \(a^{7}\). Next, find the GFC, which is \(a^{3}\). Then, divide each term by the GCF:

• \frac{a^{3}}{a^{3}}=1
• \frac{a^{4}}{a^{3}}=a
• \frac{a^{5}}{a^{3}}=a^{2}
• \frac{a^{7}}{a^{3}}=a^{4}

This simplifies the original polynomial to: \(a^{3}(1 + a + a^{2} + a^{4})\). Factoring out the GCF is a foundational skill in algebra, often simplifying complex equations into more manageable expressions.

Algebraic Expressions

Understanding algebraic expressions is key to mastering algebra. An algebraic expression is a combination of constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division. The goal is often to simplify these expressions. In the provided example, the algebraic expression was: a^{3}+a^{4}+a^{5}+a^{7}. By systematically applying factoring techniques, we transformed it into a more compact and useful form, \(a^{3}(1 + a + a^{2} + a^{4})\). Knowing how to manipulate algebraic expressions is critical for solving equations, understanding functions, and working with more advanced mathematics concepts. Always look for patterns, common factors, and opportunities to simplify when working with expressions.