Problem 49
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. $$ 8 a^{2} x^{3}-2 b^{2} x $$
Step-by-Step Solution
Verified Answer
Factored form is \(2x(2ax + b)(2ax - b)\).
1Step 1: Identify Common Factors
First, look at each term in the expression \( 8a^2x^3 - 2b^2x \). Identify the greatest common factor (GCF) of the coefficients and variables. Here, the GCF of the coefficients 8 and 2 is 2, and both terms share the variable \( x \). So, the GCF of the entire expression is \( 2x \).
2Step 2: Factor Out the GCF
Next, divide each term in the expression by the GCF and factor it out. This gives: \[ 2x(\frac{8a^2x^3}{2x} - \frac{2b^2x}{2x}) = 2x(4a^2x^2 - b^2) \] Thus, the expression is factored as \( 2x(4a^2x^2 - b^2) \).
3Step 3: Check for Further Factorization
Now, examine \( 4a^2x^2 - b^2 \) to see if it can be factored further. Recognize that this expression resembles the difference of squares \( A^2 - B^2 \), where \( A = 2ax \) and \( B = b \). Since the formula for the difference of squares is \( A^2 - B^2 = (A + B)(A - B) \), apply it to factor \( 4a^2x^2 - b^2 \).
4Step 4: Apply the Difference of Squares Formula
Using the difference of squares, factor \( 4a^2x^2 - b^2 \) as \((2ax + b)(2ax - b)\). Therefore, the complete factorization of the original expression is: \[ 2x(2ax + b)(2ax - b) \] This is the final factored form of the expression.
Key Concepts
Algebraic ExpressionsDifference of SquaresGreatest Common Factor
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They form the foundation of algebra and help in representing real-world situations mathematically. The concept of factoring is essential in simplifying these expressions and solving equations. It involves rewriting an expression as a product of its factors, making it easier to handle.
In the expression provided, each term is made up of coefficients and variables. Understanding how these components interact is crucial for successful factorization. By recognizing patterns in algebraic expressions, such as grouping like terms or identifying common factors, we can simplify complex problems effectively.
In the expression provided, each term is made up of coefficients and variables. Understanding how these components interact is crucial for successful factorization. By recognizing patterns in algebraic expressions, such as grouping like terms or identifying common factors, we can simplify complex problems effectively.
Difference of Squares
Difference of squares is a powerful tool in algebra for simplifying certain types of polynomial expressions. It refers to the subtraction of one perfect square from another, expressed as \( A^2 - B^2 \). This specific form can be factored into a product of two binomials, \( (A + B)(A - B) \).
For example, in the expression \( 4a^2x^2 - b^2 \), we identify \( A = 2ax \) and \( B = b \) because both \( 4a^2x^2 \) and \( b^2 \) are perfect squares. Applying the difference of squares rule allows us to rewrite it as \( (2ax + b)(2ax - b) \). This method is efficient and reveals deeper insights into the structure of polynomial expressions, aiding in their factorization.
For example, in the expression \( 4a^2x^2 - b^2 \), we identify \( A = 2ax \) and \( B = b \) because both \( 4a^2x^2 \) and \( b^2 \) are perfect squares. Applying the difference of squares rule allows us to rewrite it as \( (2ax + b)(2ax - b) \). This method is efficient and reveals deeper insights into the structure of polynomial expressions, aiding in their factorization.
Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor that can evenly divide each term of an algebraic expression. Finding the GCF is the first step in factoring polynomials, as it simplifies the expression and makes it more manageable.
For the expression \( 8a^2x^3 - 2b^2x \), identifying the GCF involves examining both the numerical coefficients and the variables. We find that the GCF of the coefficients 8 and 2 is 2, and the variable \( x \) is shared by both terms. Thus, the GCF is \( 2x \).
Factoring out the GCF from the original expression simplifies it to \( 2x(4a^2x^2 - b^2) \). This step not only aids in further factorization but also cleans up the expression, making it clearer and easier to work with. Recognizing and factoring out the GCF is a fundamental skill in algebra, crucial for handling more complex expressions efficiently.
For the expression \( 8a^2x^3 - 2b^2x \), identifying the GCF involves examining both the numerical coefficients and the variables. We find that the GCF of the coefficients 8 and 2 is 2, and the variable \( x \) is shared by both terms. Thus, the GCF is \( 2x \).
Factoring out the GCF from the original expression simplifies it to \( 2x(4a^2x^2 - b^2) \). This step not only aids in further factorization but also cleans up the expression, making it clearer and easier to work with. Recognizing and factoring out the GCF is a fundamental skill in algebra, crucial for handling more complex expressions efficiently.