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Chapter 5: Problem 87

Factor. $$-4 x^{2} y-4 x^{3}+24 x y^{2}$$

Short Answer

Expert verified
The factored form of the expression \(-4x^{2}y - 4x^{3} + 24xy^{2}\) is \(-4x(x + y^{2} - x^{2})\).

Step by step solution

01

Identify Common Factors

Inspect the given polynomial\( -4x^{2}y - 4x^{3} + 24xy^{2}\)and identify any common factors. Examining all three terms of the polynomial, a common factor of -4x is evident. This important step simplifies the expression for further factoring.
02

Factor Out the Common Factors

By factoring out the common term -4x from all terms, the expression is simplified to \( -4x(x + y^{2} - x^{2})\). In this step, each term in the original expression is divided by -4x. The polynomial \(x^{2}y + x^{3} - 6y^{2}\) is the result because \( -4x^{2}y/-4x = xy\), \( -4x^{3}/-4x = x^{2}\) and \(16xy^{2}/-4x = 4y^{2}\).
03

Review Factored Expression

The factored form of the given expression is the polynomial \( -4x(x + y^{2} - x^{2})\). It is important to verify the answer by distributing -4x back through the terms in the parentheses to ensure the original polynomial is obtained.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factors
When looking at a polynomial, the first step in factoring it is to search for any common factors. This means finding a number or term that can be evenly divided into each part of the polynomial. For example, in the polynomial \(-4x^{2}y - 4x^{3} + 24xy^{2}\),we need to see what is common in all the terms.
The common factor here is\(-4x\).
  • \(-4\) is common because each term is a multiple of \(-4\).
  • \(x\) is common because each term contains at least one \(x\).
Finding common factors is an important step because it simplifies the polynomial, making it easier to work with. It reduces complexity and often reveals additional factoring opportunities. It acts like pulling a thread that simplifies and clarifies the entire expression.
Factored Form
Once a common factor is identified, the next step is to write the polynomial in its factored form. Factoring means breaking the expression into multiples of simpler expressions. After extracting the common factor, the polynomial becomes more straightforward. In our example with the polynomial \(-4x^{2}y - 4x^{3} + 24xy^{2}\), when we factor out \(-4x\), we are left with:\[-4x(x + y^{2} - x^{2})\].
The goal is to simplify the expression to a product of terms, making it easier to analyze and solve, especially when dealing with equations or searching for zeros.
It's like rearranging a room. By organizing everything efficiently, you can see the entire picture clearly.
  • This expression now reveals its inner structure and dependencies between variables and constant terms.
  • This step also helps if further factorization is possible, often leading to a completely factored form.
Algebraic Expressions
Algebraic expressions consist of terms that are combined through addition, subtraction, and multiplication. These expressions can have constants, variables, and their positive powers. Understanding their structure is foundational.In the polynomial \(-4x^{2}y - 4x^{3} + 24xy^{2}\),you see all elements in action, involving variables \(x, y\)and coefficients like \(-4\)and \(24\).
  • A term like \(-4x^{2}y\)consists of a constant multiplier \(-4\), and variable parts x^{2}y\.
  • The structure of terms allows mathematicians to apply various operations and transformations, including factoring.
Algebraic expressions form the basis of equations and functions. Mastering them helps you understand more complex mathematical concepts and solve wide-ranging problems.

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Most popular questions from this chapter

Explain how you would factor \(x(a-b)+y(b-a)\).

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