Textbook Question
In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (−6x3+5x2−8x+9)+(17x3+2x2−4x−13)
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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 7
In Exercises 5–8, find the degree of the polynomial. x2−4x3+9x−12x4+63
Verified step by step guidance1
Step 1: Understand the concept of the degree of a polynomial. The degree of a polynomial is the highest power of the variable (in this case, x) with a non-zero coefficient.
Step 2: Identify all the terms in the polynomial. The given polynomial is: . The terms are: , , , , and .
Step 3: Determine the degree of each term. The degree of a term is the exponent of the variable x. For the terms: has degree 2, has degree 3, has degree 1, has degree 4, and (a constant) has degree 0.
Step 4: Identify the highest degree among all the terms. From the degrees calculated in Step 3, the highest degree is 4, which comes from the term .
Step 5: Conclude that the degree of the polynomial is 4, as it is the highest power of x with a non-zero coefficient.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the polynomial's behavior as the variable approaches infinity and helps determine the number of roots and the shape of its graph. For example, in the polynomial x^2 - 4x^3 + 9x - 12x^4 + 63, the degree is determined by the term with the highest exponent.
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Standard Form of Polynomials
Polynomial Terms
A polynomial is composed of terms, which are individual components that can include constants, variables, and exponents. Each term is typically in the form of ax^n, where 'a' is a coefficient, 'x' is the variable, and 'n' is a non-negative integer. Understanding how to identify and classify these terms is essential for determining the degree of the polynomial.
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Standard Form of a Polynomial
The standard form of a polynomial arranges its terms in descending order of their exponents. This format makes it easier to identify the leading term and the degree of the polynomial. For instance, rewriting the polynomial x^2 - 4x^3 + 9x - 12x^4 + 63 in standard form would help quickly identify the term with the highest degree, which is crucial for solving the problem.
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Related Practice
Textbook Question
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x2-6x+3, for x=7
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Textbook Question
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √144+25
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Textbook Question
In Exercises 1–10, factor out the greatest common factor. x(2x+1)+4(2x+1)
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Textbook Question
Evaluate each exponential expression in Exercises 1–22. (−3)0
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Textbook Question
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (3x−9)/(x2−6x+9)
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