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Chapter 0: Problem 22

Determine the degree of the polynomial. $$ -4.7 a b c^{4}-5.2 a^{2} b c^{5}+2.6 a^{3} c $$

Short Answer

Expert verified
The degree of the polynomial is 8.

Step by step solution

01

- Identify the terms

The polynomial consists of three terms: 1. -4.7 abc^{4}2. -5.2 a^{2}bc^{5}3. 2.6 a^{3}c
02

- Find the degree of each term

For the term -4.7 abc^{4}, the degree is 1 + 1 + 4 = 6. For the term -5.2 a^{2} bc^{5}, the degree is 2 + 1 + 5 = 8. For the term 2.6 a^{3} c, the degree is 3 + 1 = 4.
03

- Determine the highest degree

The degrees of the terms are 6, 8, and 4. The highest degree is 8.

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Key Concepts

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

polynomial terms identification
To begin with, let's break down what a polynomial is. A polynomial is a mathematical expression consisting of variables (sometimes called indeterminates), coefficients, and non-negative integer exponents. Each part of the polynomial separated by a plus (+) or minus (-) sign is called a 'term'. For example, in the expression: -4.7 abc^{4} - 5.2 a^{2}bc^{5} + 2.6 a^{3} c, we have three terms:
  • -4.7 abc^{4}
  • -5.2 a^{2}bc^{5}
  • 2.6 a^{3} c
Identifying these terms accurately is the foundation of determining the polynomial's degree.
degree of polynomial terms
After identifying each term, the next step is to determine the degree of each term. The degree of a term in a polynomial is the sum of the exponents of the variables in that term. Let's break this down for our given terms: For the term -4.7 abc^{4}:
  • The variable a has an exponent of 1.
  • The variable b has an exponent of 1.
  • The variable c has an exponent of 4.
Adding these exponents together, we get: 1 + 1 + 4 = 6. Therefore, the degree of the term -4.7 abc^{4} is 6. Now, for the term -5.2 a^{2}bc^{5}:
  • The variable a has an exponent of 2.
  • The variable b has an exponent of 1.
  • The variable c has an exponent of 5.
Adding these, we get: 2 + 1 + 5 = 8. Hence, the degree of the term -5.2 a^{2}bc^{5} is 8. Lastly, for the term 2.6 a^{3} c:
  • The variable a has an exponent of 3.
  • The variable c has an exponent of 1.
Summing them gives: 3 + 1 = 4. So, the degree of the term 2.6 a^{3} c is 4.
highest degree in a polynomial
Once we have the degrees of all individual terms within the polynomial, the final step is to determine the highest degree among them. This highest degree is what we refer to as the degree of the polynomial. From our calculations, we have the following degrees:
  • The degree of the term -4.7 abc^{4} is 6.
  • The degree of the term -5.2 a^{2}bc^{5} is 8.
  • The degree of the term 2.6 a^{3} c is 4.
Among these, the highest degree is 8. Hence, the degree of the polynomial -4.7 abc^{4} - 5.2 a^{2}bc^{5} + 2.6 a^{3} c is 8. This means that 8 is the highest sum of the exponents on the variables in any individual term of the polynomial.

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Most popular questions from this chapter

A polynomial in the variable \(x\) is defined as an expression of the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0} .\) Explain what this means.

Simplify each expression. $$ (x-4)^{2}-6(x+1)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{5+2 \sqrt{x}})(\sqrt{5-2 \sqrt{x}}) $$

Determine the sign of the expression. Assume that \(a, b,\) and \(c\) are real numbers and \(a<0, b>0,\) and \(c<0\). $$\frac{a b^{2}}{c^{3}}$$

Determine if the statement is true or false. The sum of two polynomials each of degree 5 will be less than or equal to degree 5 .
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