Textbook Question
Factor completely, or state that the polynomial is prime.
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 65
In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. (3x4 y2+5x3 y−3y)−(2x4 y2−3x3 y−4y+6x)
Verified step by step guidance1
Start by distributing the negative sign across the second polynomial. This means you will change the sign of each term in the second polynomial: \((2x^4 y^2 - 3x^3 y - 4y + 6x)\) becomes \(-2x^4 y^2 + 3x^3 y + 4y - 6x\).
Rewrite the expression by combining the first polynomial and the modified second polynomial: \((3x^4 y^2 + 5x^3 y - 3y) + (-2x^4 y^2 + 3x^3 y + 4y - 6x)\).
Group like terms together. Like terms are terms that have the same variables raised to the same powers. For example, group \(3x^4 y^2\) with \(-2x^4 y^2\), \(5x^3 y\) with \(3x^3 y\), \(-3y\) with \(4y\), and \(-6x\) remains as is.
Combine the coefficients of the like terms. For example, \(3x^4 y^2 - 2x^4 y^2\), \(5x^3 y + 3x^3 y\), \(-3y + 4y\), and \(-6x\).
After simplifying, identify the degree of the resulting polynomial. The degree of a polynomial is determined by the term with the highest sum of the exponents of its variables. For example, in \(x^4 y^2\), the degree is \(4 + 2 = 6\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Operations
Polynomial operations include addition, subtraction, multiplication, and division of polynomials. In this context, we are focusing on subtraction, which involves combining like terms from two polynomials. Understanding how to identify and combine like terms is crucial for simplifying the expression correctly.
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Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the polynomial expression. It provides insight into the polynomial's behavior and shape. After performing operations on polynomials, determining the degree helps classify the resulting polynomial and understand its characteristics.
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Like Terms
Like terms are terms in a polynomial that have the same variable raised to the same power. For example, in the expression 3x^2 and 5x^2, both terms are like terms because they share the same variable and exponent. Recognizing and combining like terms is essential for simplifying polynomials and performing operations accurately.
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Related Practice
Textbook Question
In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125
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Textbook Question
In Exercises 33–68, add or subtract as indicated. x/(x2−2x−24) − x/(x2−7x+6)
Textbook Question
Evaluate each expression 1251/3.
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Textbook Question
Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number.
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Textbook Question
Write each number in decimal notation without the use of exponents.