Problem 1 Is the given expression a polyno... [FREE SOLUTION]

Chapter 6: Problem 1

Is the given expression a polynomial? Why or why not? $$-5 z^{2}-4 z+12$$

Short Answer

Expert verified

Yes, the given expression $-5 z^{2}-4 z+12$ is a polynomial because it consists of a sum or difference of terms, each term is a multiple of a power of the variable z, and the exponents on the variable z are non-negative whole numbers.

Step by step solution

Identify the terms of the expression

The expression is given as $-5 z^{2}-4 z+12$. It consists of three terms separated by addition or subtraction operations: $-5 z^{2}$, $-4 z$, and $12$.

Check if the terms are multiples of a power of the variable

All three terms are multiples of a power of the variable z: - The first term, $-5 z^{2}$, is a multiple of $z^{2}$. - The second term, $-4 z$, is a multiple of $z$. - The third term, $12$, is a multiple of $z^{0}$ (since any non-zero number to the power of 0 is 1).

Verify the exponents on the variable are non-negative whole numbers

Let's examine the powers of z in each term: - In the first term, the exponent on z is 2 (a non-negative whole number). - In the second term, the exponent on z is 1 (a non-negative whole number). - In the third term, the exponent on z is 0 (when considered as a multiple of z with an implicit exponent of 0, also a non-negative whole number). Since the expression $-5 z^{2}-4 z+12$ meets all the criteria for being a polynomial, as stated in the analysis, we can conclude that the given expression is indeed a polynomial.

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Key Concepts

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

Terms in an Expression

In mathematics, an expression is essentially a combination of numbers, variables, and operators like addition, subtraction, etc. To understand an expression, it's important to look at its parts known as "terms." Terms are the building blocks of an expression. They are separated by addition or subtraction signs. In the expression \[-5 z^{2} - 4 z + 12\], you can see that there are three terms:

-5 z^{2}
-4 z
12

Each term can be made up of a coefficient and a variable raised to a power. For example, in the term $-5 z^{2}$, "-5" is the coefficient, and $z^{2}$ is the variable raised to the power of 2. Understanding terms is crucial because they help identify whether the expression is a polynomial.

Non-negative Exponents

Exponents determine the power to which a number or a variable is raised. When it comes to polynomials, the exponents of the variables must be non-negative whole numbers. This means they can be 0, 1, 2, 3, and so on, but not negative or fractional.
Let's look at our example expression. The exponents on each term of $-5 z^{2} - 4 z + 12$ are:

2 in $-5 z^{2}$
1 in $-4 z$
0 in $12$ (since $z^{0} = 1$).

All these exponents are non-negative whole numbers. This is an essential characteristic of polynomials. Non-negative exponents ensure the resulting expressions have a certain behavior and are functions with a degree, which allows polynomials to be used widely in mathematics and applications.

Understanding Variable Powers

Variable powers in polynomials determine the degree of each term and the entire expression. They indicate how many times the variable is used as a factor. For a polynomial to be valid, these powers must adhere to specific rules.
In our expression $-5 z^{2} - 4 z + 12$, checking the variable powers involves:

First term: $z^{2}$ which is a square, thus a power of 2.
Second term: $z$ which is implicitly $z^{1}$.
Third term: The constant $12$ which is like $z^{0}$.

Variable powers can impact the function's graph, influence calculations, and determine the degree of the polynomial as the highest variable power. This degree impacts complexity but also functionality, such as in solving equations and modeling real-world problems.

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91影视

Is the given expression a polynomial? Why or why not? $$-5 z^{2}-4 z+12$$

Short Answer

Step by step solution

Identify the terms of the expression

Check if the terms are multiples of a power of the variable

Verify the exponents on the variable are non-negative whole numbers

Key Concepts

Terms in an Expression

Non-negative Exponents

Understanding Variable Powers

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