Problem 47 Identify the degree of each poly... [FREE SOLUTION]

Chapter 3: Problem 47

Identify the degree of each polynomial. $$19 t^{9}+4 t^{3}+25 t^{12}-11 t^{5}$$

Short Answer

Expert verified

Answer: The degree of the given polynomial is 12.

Step by step solution

Identify the terms in the polynomial

The given polynomial is: $$19 t^{9}+4 t^{3}+25 t^{12}-11 t^{5}$$ Notice that there are four terms in the polynomial: $19 t^{9}, 4 t^{3}, 25 t^{12}, -11 t^{5}$.

Identify the exponent of each term

Examine each term and identify the exponent (the power of the variable $t$). The exponents are as follows: - For $19 t^{9}$, the exponent is 9. - For $4 t^{3}$, the exponent is 3. - For $25 t^{12}$, the exponent is 12. - For $-11 t^{5}$, the exponent is 5.

Find the highest exponent

Compare the exponents and find the highest exponent. In this case, the highest exponent is 12 from the term $25 t^{12}$.

State the degree of the polynomial

Since the highest exponent (power of the variable $t$) is 12, the degree of the given polynomial is 12.

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Key Concepts

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

Polynomials

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents combined using addition, subtraction, and multiplication but not division by a variable. Polynomials are fundamental in algebra and appear in various forms and degrees. Here are some key features of polynomials:

A polynomial can have constants, variables like $x$ or $t$, and numeric coefficients, such as $-11$ or $19$.
The terms in a polynomial are separated by addition (+) or subtraction (-) signs.
Each term is a product of a constant and a whole number power of the variable, such as $4t^3$.

Polynomials provide the foundation for more complex algebraic operations and are used in various fields such as physics, engineering, and economics.

Exponents

Exponents in polynomials indicate how many times a variable is multiplied by itself. They play a crucial role in determining the characteristics and behavior of the polynomial.

The exponent is a small number written above and to the right of the variable (e.g., $t^9$ means "$t$ to the ninth power").
Exponents must be whole numbers in a polynomial.
Knowing the exponents of a polynomial helps you identify its degree, which is a primary feature that describes the polynomial's complexity.

Understanding exponents makes it easier to grasp how polynomials behave and change as variables are altered.

Terms of a Polynomial

The terms of a polynomial are the distinct parts separated by addition or subtraction. Each term includes a coefficient, a variable, and an exponent:

Coefficient: This is the numeric factor in the term. For example, in $19t^9$, 19 is the coefficient.
Variable: The letter representing the unknown value, like $t$ in our example.
Exponent: The power to which the variable is raised; here, 9 in $t^9$.

By examining each term in a polynomial, you can determine the polynomial's degree and how complicated the function is. Comprehensive understanding of terms aids in simplifying, solving, and graphing polynomials effectively.

Highest Degree Term

In a polynomial, the degree is determined by the highest exponent among its terms. The term with the largest exponent is known as the highest degree term.

The degree tells you the most significant power in the polynomial, indicating its overall behavior and growth.
For example, in the polynomial $19t^9 + 4t^3 + 25t^{12} - 11t^5$, the term $25t^{12}$ has the highest exponent of 12.
The highest degree term often drives the long-term behavior of the polynomial in terms of its graph and solutions.

Long polynomials may have many terms, but identifying the highest degree term quickly clarifies the polynomial's essential characteristics.

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91影视

Identify the degree of each polynomial. $$19 t^{9}+4 t^{3}+25 t^{12}-11 t^{5}$$

Short Answer

Step by step solution

Identify the terms in the polynomial

Identify the exponent of each term

Find the highest exponent

State the degree of the polynomial

Key Concepts

Polynomials

Exponents

Terms of a Polynomial

Highest Degree Term

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