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Chapter 5: Problem 16

Identify the terms of each polynomial. \(5 a^{3}+4 a^{2}-a-7\)

Short Answer

Expert verified
The terms are \(5 a^{3}\), \(4 a^{2}\), \(-a\), and \(-7\).

Step by step solution

01

Understand the Definition of a Term

A term in a polynomial is a combination of variables, coefficients, and sometimes constants, separated by addition or subtraction operations.
02

Identify and List Each Term

Examine the polynomial and split it into individual terms. The polynomial is given as: \[5 a^{3} + 4 a^{2} - a - 7\] The terms in this polynomial are separated by addition (+) or subtraction (-) signs.
03

Write Down Each Term

List each term separately: 1. \(5 a^{3}\) 2. \(4 a^{2}\) 3. \(-a\) 4. \(-7\) These are the individual terms of the polynomial.

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Key Concepts

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

polynomial
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it consists of terms that are added or subtracted from each other. Polynomials can have various degrees based on the highest power (exponent) of the variable present in the expression. For example, in the polynomial \(5a^3 + 4a^2 - a - 7\), the highest exponent is 3, making it a cubic polynomial. Remember, polynomials are important in algebra and serve as the foundation for many mathematical concepts.
terms of a polynomial
The terms of a polynomial are the individual parts separated by addition or subtraction signs. Each term can be composed of variables, coefficients, and sometimes constants. For the polynomial given in the exercise \(5a^3 + 4a^2 - a - 7\), there are four terms:
  • \(5a^3\)
  • \(4a^2\)
  • \(-a\)
  • \(-7\)
Understanding how to identify and separate these terms is crucial. Each term stands independently with its own coefficient and variable powers.
variables and coefficients
Variables and coefficients are key components of polynomial terms. A variable is a symbol (like a, x, y) that represents a number we often do not know yet. A coefficient is a numerical factor that multiplies the variable. In the term \(5a^3\), '5' is the coefficient and 'a' is the variable raised to the power of 3. Similarly, in \(-a\), the coefficient is implicitly \(-1\) since any variable without a visible coefficient has one implicitly present. Recognizing variables and coefficients helps in understanding the behavior of each term in the polynomial.
constants in polynomials
Constants in polynomials are terms that do not contain any variables. They are simply fixed numbers. For instance, in the polynomial \(5a^3 + 4a^2 - a - 7\), the term \(-7\) is a constant because it does not change with the variable. Constants play a significant role in shifting the value of the polynomial up or down on a graph, but they do not affect the overall shape. Recognizing constants helps understand how polynomials are structured and how they behave.

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Most popular questions from this chapter

Solve. Write the answers using scientific notation. Without performing actual computations, explain why \(3^{-29}\) is smaller than \(2^{-29}\)

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{5 a^{7}}{2 b^{5} c}\right)^{0} $$

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

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=3 x^{2}\\\ &g(x)=\frac{1}{x-9} \end{aligned} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -8(-10) $$
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