Problem 91
Question
Factor the expression completely. (This type of expression arises in calculus in using the “product rule.”) $$ 3 x^{2}(4 x-12)^{2}+x^{3}(2)(4 x-12)(4) $$
Step-by-Step Solution
Verified Answer
The completely factored expression is \(4x^2(4x - 12)(5x - 9)\).
1Step 1: Identify common factors
First, we need to identify common factors among all terms in the expression. Both terms have a common factor of \(x^2\) and \((4x - 12)\). Factoring out \(x^2(4x - 12)\), we start simplifying the expression.
2Step 2: Factor out the greatest common factor
Take the factors \(x^2\) and \((4x - 12)\) out of the expression. This leaves us with: \[x^2 (4x-12) (3(4x-12) + 8x)\].
3Step 3: Simplify inside the parentheses
Now, we simplify the expression inside the parentheses: \[3(4x - 12) + 8x = 12x - 36 + 8x = 20x - 36\].
4Step 4: Factor the simplified expression inside the parentheses
The expression \(20x - 36\) can be factored by taking out the greatest common factor, which is 4. This gives us: \[4(5x - 9)\].
5Step 5: Write the complete factored expression
Substitute the simplified factored expression back into the full expression: \[x^2 (4x-12) 4(5x-9)\].
6Step 6: Simplify the expression further
Simplify the expression by combining factors: \[4x^2(4x - 12)(5x - 9)\].
Key Concepts
Common FactorGreatest Common FactorProduct RuleSimplifying Expressions
Common Factor
When working with algebraic expressions, a common factor refers to a number or variable that can divide each term of an expression evenly. In other words, it's something shared that can be "factored out" to simplify the expression. Consider the exercise provided. Here, both terms in the expression share certain factors. They both include a factor of \(x^2\) and \((4x - 12)\).
By identifying and factoring out these common factors, we can rewrite the expression in a more simplified form, which is the critical first step in making the expression easier to work with. Factoring common elements from an expression reduces complexity and often transforms a challenging algebra problem into something more manageable.
Finding and extracting common factors are essential steps that set the stage for further simplification and solving processes. A key tip is to always look for common numerical coefficients or variables present across all terms for ease in further manipulation.
By identifying and factoring out these common factors, we can rewrite the expression in a more simplified form, which is the critical first step in making the expression easier to work with. Factoring common elements from an expression reduces complexity and often transforms a challenging algebra problem into something more manageable.
Finding and extracting common factors are essential steps that set the stage for further simplification and solving processes. A key tip is to always look for common numerical coefficients or variables present across all terms for ease in further manipulation.
Greatest Common Factor
The greatest common factor (GCF) is a number or variable that can be multiplied to create all terms of a given expression, representing the largest shared factor. In the factoring process, identifying the GCF allows us to reduce an expression to its simplest form more efficiently.
In the given exercise, the GCF in each term includes the expression \((4x - 12)\). But it's not just any common factor; it's the largest one, which means it incorporates all the factors common to both terms. This ensures that once it's factored out, the resulting expression has the simplest form possible and can be dealt with more easily in subsequent steps.
By effectively factoring out the GCF, like \(x^2\) and \((4x - 12)\), we clear out the expression, laying a straightforward foundation for further simplification and solving. It ensures that any remaining operations or factorizations are as concise as possible.
In the given exercise, the GCF in each term includes the expression \((4x - 12)\). But it's not just any common factor; it's the largest one, which means it incorporates all the factors common to both terms. This ensures that once it's factored out, the resulting expression has the simplest form possible and can be dealt with more easily in subsequent steps.
By effectively factoring out the GCF, like \(x^2\) and \((4x - 12)\), we clear out the expression, laying a straightforward foundation for further simplification and solving. It ensures that any remaining operations or factorizations are as concise as possible.
Product Rule
The product rule is a concept from calculus often used to differentiate products of two or more functions. It states that the derivative of a product of functions is the sum of the derivative of each individual function multiplied by the others. Although primarily a calculus concept, the algebraic manipulation related to it can come up in exercises like this.
This specific expression in the original problem can arise when using the product rule, showcasing its importance in organizing terms for calculus problems. Algebraic factorization simplifies such expressions before applying calculus rules, making them easier to handle.
For example, breaking down the product \(3x^2(4x-12)^2 + x^3(2)(4x-12)(4)\) into simpler components helps clarify its underlying structure. Thus, understanding the product rule indirectly benefits through its connection with algebraic structure, allowing you to manage complex products skillfully.
This specific expression in the original problem can arise when using the product rule, showcasing its importance in organizing terms for calculus problems. Algebraic factorization simplifies such expressions before applying calculus rules, making them easier to handle.
For example, breaking down the product \(3x^2(4x-12)^2 + x^3(2)(4x-12)(4)\) into simpler components helps clarify its underlying structure. Thus, understanding the product rule indirectly benefits through its connection with algebraic structure, allowing you to manage complex products skillfully.
Simplifying Expressions
Simplifying expressions in algebra is the process of reducing a complex expression to its most compact, efficient, and readable form. It involves combining like terms, factoring, and performing basic arithmetic operations.
In the original step-by-step solution, after identifying and removing common factors and the GCF, the exercise guides us through simplifying the remaining expression within parentheses. This process not only simplifies the expression but also makes it easier to understand and work with.
Continual refinements of simplification occur until you achieve an expression that's easier to interpret, compute, or apply further in problems like solving equations. This decreased complexity helps when these expressions are later used in calculus, where neat expressions are crucial to avoid errors.
In the original step-by-step solution, after identifying and removing common factors and the GCF, the exercise guides us through simplifying the remaining expression within parentheses. This process not only simplifies the expression but also makes it easier to understand and work with.
Continual refinements of simplification occur until you achieve an expression that's easier to interpret, compute, or apply further in problems like solving equations. This decreased complexity helps when these expressions are later used in calculus, where neat expressions are crucial to avoid errors.
- Combine like terms
- Factor completely
- Find and use the GCF
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