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Chapter 6: Problem 28

Simplify. $$ 2 x y\left(3 x y^{3}-4 x y+2 y^{4}\right) $$

Short Answer

Expert verified
The simplified expression is \(6x^2y^4 - 8x^2y^2 + 4xy^5\).

Step by step solution

01

Distribute the Terms

To simplify the expression, distribute the term outside the parenthesis, \(2xy\), to each term inside the parenthesis, \(3xy^3 - 4xy + 2y^4\). This yields:1. \(2xy \cdot 3xy^3 = 6x^2y^4\)2. \(2xy \cdot -4xy = -8x^2y^2\)3. \(2xy \cdot 2y^4 = 4xy^5\)
02

Write the Simplified Expression

Combine the distributed terms into a single expression to complete the simplification. The expression now becomes:\[6x^2y^4 - 8x^2y^2 + 4xy^5\]

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Key Concepts

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

Distributive Property
The distributive property is a fundamental tool in algebra, particularly useful for simplifying expressions and equations. It allows us to multiply a term outside a parenthesis by each term inside, spreading the multiplication across the addition or subtraction inside. For the expression given, we apply the distributive property by taking the term outside, which is \(2xy\), and multiplying it with each term inside the parenthesis: \(3xy^3\), \(-4xy\), and \(2y^4\).
  • For \(2xy \cdot 3xy^3\), you multiply the numerical coefficients (2 and 3) to get 6, then apply multiplication of powers for variables \(x\) and \(y\). This results in \(6x^2y^4\).
  • Next, \(2xy \cdot -4xy\) gives us \(-8x^2y^2\), following similar steps with coefficients and powers.
  • Finally, \(2xy \cdot 2y^4\) results in \(4xy^5\), keeping the \(x\)'s power unchanged while adding the powers of \(y\).
The distributive property is crucial when managing expressions with multiple terms and variables, simplifying into manageable equations.
Polynomials
Polynomials are mathematical expressions that consist of variables and coefficients, structured into terms connected by addition or subtraction. Each term typically includes a variable raised to a power (the exponent), multiplied by a numerical coefficient. Understanding polynomials involves recognizing these components and knowing how to manipulate them.
  • In our example, terms like \(3xy^3\), \(-4xy\), and \(2y^4\) form a polynomial inside the parenthesis. Each term has its own set of coefficients and variables.
  • A key aspect of working with polynomials is the ability to reorganize and combine like terms to simplify the expression further, which is often necessary after distributing multiplications.
  • Polynomials of different degrees involve different powers of variables; here, powers of \(y\) vary from 2 to 5, showing the polynomial nature.
Handling polynomials efficiently requires recognizing patterns and using algebraic rules, ensuring each term is appropriately managed.
Multiplication of Variables
Multiplying variables in algebra follows straightforward rules, especially when dealing with powers and exponents. When you multiply variables, each with an exponent, you add the exponents of like bases, following the law \(a^m \cdot a^n = a^{m+n}\). This principle simplifies what might initially seem complicated.
  • For instance, in \(2xy \cdot 3xy^3\), you multiply the coefficients (2 and 3), then apply the law of exponents: the \(x\)'s exponents \((1+1)\) yield \(x^2\), while \(y\)'s \((1+3)\) yield \(y^4\).
  • This step-by-step approach ensures each variable is correctly accounted for, without altering the overall operation improperly.
Understanding multiplication of variables not only aids in simplifying expressions but also instills deeper comprehension of algebra's foundations, empowering further exploration and manipulation of algebraic expressions.

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Most popular questions from this chapter

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{2}+5 x+7\right) \div(x+k) $$

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{3}-4 x^{2}+2 x-1 $$

ENGINEERING. For Exercises 32 and \(33,\) use the following information. When a certain type of plastic is cut into sections, the length of each section determines its strength. The function \(f(x)=x^{4}-14 x^{3}+69 x^{2}-140 x+100\) can describe the relative strength of a section of length \(x\) feet. Sections of plastic \(x\) feet long, where \(f(x)=0,\) are extremely weak. After testing the plastic, engineers discovered that sections 5 feet long were extremely weak. Are there other lengths of plastic that are extremely weak? Explain your reasoning.

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{3}+4 x^{2}+x+k\right) \div(x+2) $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. \(3 x^{4}-21 x^{3}+38 x^{2}-14 x+24 ; x-3\)
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