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Chapter 5: Problem 24

Determine the coefficient and the degree of term in polynomial. \(3 a^{5}-a^{3}+a-9\)

Short Answer

Expert verified
Coefficients: 3, -1, 1, -9. Degrees: 5, 3, 1, 0.

Step by step solution

01

Identify each term in the polynomial

The given polynomial is \(3a^5 - a^3 + a - 9\). Identify each term separately: \(3a^5\), \(-a^3\), \(a\), and \(-9\).
02

Determine the coefficient of each term

For each term, identify the coefficient:1. The term \(3a^5\): the coefficient is 3.2. The term \(-a^3\): the coefficient is -1.3. The term \(a\): the coefficient is 1 (since \(a\) is equivalent to \(1a\)).4. The constant term \(-9\) has no variable and its 'coefficient' is considered as -9.
03

Determine the degree of each term

For each term, identify the degree, which is the exponent of the variable a:1. The term \(3a^5\): the degree is 5.2. The term \(-a^3\): the degree is 3.3. The term \(a\): the degree is 1 (since \(a\) is equivalent to \(a^1\)).4. The constant term \(-9\) has a degree of 0 as it has no variable.

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Key Concepts

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

Understanding Polynomial Terms
A polynomial is made up of several terms, which are individual parts separated by plus or minus signs. For example, in the polynomial \(3a^5 - a^3 + a - 9\), we have four terms: \(3a^5\), \(-a^3\), \(a\), and \(-9\). Each term can consist of a coefficient and a variable raised to an exponent. In this case, the variable is \(a\). When identifying terms, it鈥檚 important to separate and observe each part of the polynomial.
Identifying Coefficients in Polynomials
The coefficient in a polynomial term is the numerical factor that multiplies the variable. In the polynomial \(3a^5 - a^3 + a - 9\), each term has a specific coefficient:
  • The term \(3a^5\) has a coefficient of 3.
  • The term \(-a^3\) has a coefficient of -1 (since the variable is multiplied by -1).
  • The term \(a\) or \(1a\) has a coefficient of 1.
  • The constant term \(-9\) is unique because it does not have a variable and its coefficient is simply -9.
The coefficients are vital as they determine the numerical value of each term when the variable is given a number.
Determining the Degree of a Polynomial
The degree of a polynomial term is the exponent to which the variable is raised. It's a crucial characteristic because it indicates the polynomial's highest power. For our polynomial \(3a^5 - a^3 + a - 9\), we determine the degrees of each term as follows:
  • The term \(3a^5\) has a degree of 5 because the variable \(a\) is raised to the power of 5.
  • The term \(-a^3\) has a degree of 3.
  • The term \(a\) is equivalent to \(a^1\), thus has a degree of 1.
  • The constant term \(-9\) does not have a variable and is considered to have a degree of 0.
The degree of the entire polynomial is the highest degree of any term within it. In this case, the degree of the polynomial \(3a^5 - a^3 + a - 9\) is 5, the highest degree among its terms.

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Most popular questions from this chapter

Simplify. Write \(81^{3} \cdot 27 \div 9^{2}\) as a power of 3

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graph to check each prediction.

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{5 x^{7} y}{-2 z^{4}}\right)^{3} $$

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Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{26} $$
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