Problem 17 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 8 p^{3} q^{7}+4 p^{2} q^{3} $$

Step-by-Step Solution

Answer

The factorization is \(4p^2q^3(2pq^4 + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at each term in the expression. We have two terms: \(8p^3q^7\) and \(4p^2q^3\). Identify the greatest common factor of the numerical coefficients, 8 and 4, which is 4. Now, find the common variable factors with the lowest powers. For \(p\), it is \(p^2\) (the lowest power present in each term), and for \(q\), it is \(q^3\). Thus, the GCF is \(4p^2q^3\).

Step 2: Factor the GCF Out

Divide each term by the GCF \(4p^2q^3\). For the first term \(\frac{8p^3q^7}{4p^2q^3} \), simplify to get \(2pq^4\). For the second term \(\frac{4p^2q^3}{4p^2q^3} \), simplify to get 1. Write the expression as a product of the GCF and the simplified terms: \[4p^2q^3(2pq^4 + 1)\]

Step 3: Verify the Factorization

Expand the factored expression \(4p^2q^3(2pq^4 + 1)\). Distribute to verify the original expression: \[4p^2q^3 \times 2pq^4 = 8p^3q^7\] \[4p^2q^3 \times 1 = 4p^2q^3\]Adding these gives the original expression \(8p^3q^7 + 4p^2q^3\). The expression is correctly factored.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is key when dealing with polynomials. The GCF involves identifying the largest factor that can divide each term in the expression without leaving a remainder. For numerical coefficients like 8 and 4, the GCF would be 4, since it's the largest number that can divide both. For variables with exponents, such as in the terms \(p^3\) and \(p^2\), you choose the smallest power, which in this instance is \(p^2\). Similarly, for \(q^7\) and \(q^3\), the GCF is \(q^3\). By combining these factors together, you get \(4p^2q^3\) as the GCF for both terms. This systematic approach ensures clarity and precision when simplifying expression components.

Polynomial Expressions

A polynomial expression is a collection of terms constituted by variables and coefficients, combined using addition, subtraction, or multiplication. Each term in a polynomial can have constants (numerical coefficients) and variables (like \(p\) and \(q\)) raised to various powers. Understanding how to work with these expressions is crucial for algebraic simplification.

• Each part of the expression, like \(8p^3q^7\), is called a term.
• The degree of the term is determined by the sum of the exponents of its variables. For example, the degree of \(p^3q^7\) is 10 (3 from \(p^3\) and 7 from \(q^7\)).
• Polynomial expressions can be simplified by factoring out common terms, which reduces complexity.

The processes involved are fundamental in resolving algebraic equations effectively.

Algebraic Simplification

The process of algebraic simplification revolves around making expressions as concise as possible without altering their value. In terms of factoring, this involves removing common factors from numerous terms. By doing this, you're streamlining the equation, making it easier to work with in further operations.

• Simplification helps us understand the underlying structure of an expression.
• By dividing each term by the GCF, what's left is a simplified product that is aesthetically and functionally reduced.
• For example, simplifying \(8p^3q^7 + 4p^2q^3\) first involves factoring out \(4p^2q^3\), leading to \(4p^2q^3(2pq^4 + 1)\), a more manageable form.

The resulting equation reveals the relationship between its components more clearly, highlighting the advantage of algebraic simplification in polynomials.