Problem 40 - 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. $$ 70 p^{4} q^{3}-35 p^{4} q^{2}+49 p^{5} q^{2} $$

Step-by-Step Solution

Answer

The expression factors to \( 7p^4q^2 (10q - 5 + 7p) \).

Step 1: Identify the Greatest Common Factor (GCF)

To start factoring the expression, identify the GCF of the coefficients and the smallest power of each variable shared by all terms. The coefficients are 70, 35, and 49, and they share a GCF of 7. For the variables, each term contains at least \( p^4 \) and \( q^2 \). Thus, the GCF of the whole expression is \( 7p^4q^2 \).

Step 2: Factor out the GCF

Factor \( 7p^4q^2 \) out of each term in the expression.\[ 70p^4q^3 = 7p^4q^2 imes 10q \]\[ -35p^4q^2 = 7p^4q^2 imes (-5) \]\[ 49p^5q^2 = 7p^4q^2 imes 7p \]This adjusts the original expression to \[ 7p^4q^2 (10q - 5 + 7p) \].

Step 3: Verify if further factoring is possible

Check if the expression inside the parentheses \( 10q - 5 + 7p \) can be factored further. Since there are no common factors remaining for this expression and it doesn't fit a special factoring pattern (such as a trinomial square or difference of squares), it remains as is.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial concept in algebra, especially when it comes to factoring polynomial expressions. In this context, the GCF refers to the largest factor that can be evenly divided from each term in a given polynomial. This includes both numerical coefficients and variable parts.To identify the GCF, you must:

• Determine the GCF of the numerical coefficients of the terms.
• Find the smallest power of each variable that appears in all terms of the expression.

Let's look at the example from the exercise. The coefficients for the terms are 70, 35, and 49. Using basic number theory, the largest number that can divide all three is 7.For the variables, all terms include at least \( p^4 \) and \( q^2 \), so we select these as part of the GCF.Thus, the GCF of the expression is \( 7p^4q^2 \).
By factoring out the GCF, the expression is simplified, making the next steps of the factoring process more manageable.

Polynomial Expressions

Polynomial expressions are mathematical phrases that can involve numbers, variables (like \( p \) and \( q \)), and operations such as addition, subtraction, and multiplication.
Polynomials can have several terms, each term being a product of a constant and a variable raised to an exponent. In the given example, the expression consists of:

• A first term: \( 70p^4q^3 \)
• A second term: \( -35p^4q^2 \)
• A third term: \( 49p^5q^2 \)

Understanding how to manage these terms effectively is key to simplifying and solving polynomial problems. Factoring is a critical technique for breaking down these expressions into more manageable pieces, as reflected in this exercise. Accurate handling of polynomial expressions notably includes recognizing the order of operations and maintaining careful attention to the variable exponents.

Factorization Techniques

Factorization techniques involve strategies for simplifying polynomials by rewriting them as products of simpler polynomials or factors.
One basic technique is to start with the GCF, which simplifies the expression right away by removing common threads throughout the terms.Once the GCF is factored out, it's important to examine whether further factoring opportunities exist. In this exercise, after factoring out \( 7p^4q^2 \), we are left with the simplified expression \( 7p^4q^2(10q - 5 + 7p) \).The remaining polynomial inside the parentheses, \(10q - 5 + 7p\), is then checked for further factorization. However, since it does not fit any specific patterns known for further breakdown, such as a difference of squares or binomial squares, it stands as its simplest form.
Students are encouraged to practice identifying these patterns and becoming familiar with them, as they greatly assist in recognizing when a polynomial is fully factored.