Problem 3

Question

For Problems $1-8$, determine the degree of each polynomial. $$ 5 x^{2}-9 $$

Step-by-Step Solution

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Answer

The degree of the polynomial is 2.

1Step 1: Identify the terms

The polynomial given is $5x^2 - 9$. This polynomial consists of two terms: $5x^2$ and $-9$.

2Step 2: Find the degree of each term

The degree of a term in a polynomial is the sum of the exponents on the variables within that term. In $5x^2$, the exponent of $x$ is 2, so the degree of $5x^2$ is 2.

3Step 3: Determine the degree of the polynomial

The degree of a polynomial is the highest degree among its terms. Since the term $5x^2$ has the highest degree, which is 2, the degree of the polynomial $5x^2 - 9$ is 2.

Key Concepts

Polynomial TermsDegree of a TermExponents in Polynomials

Polynomial Terms

In any given polynomial, the expression is made up of individual parts known as terms. Each term can include constants (numbers), variables (often represented by letters such as $x$), and exponents (power to which the variable is raised). For example, in the polynomial $5x^2 - 9$, there are two distinct terms: $5x^2$ and $-9$. When identifying terms in a polynomial:

Terms are separated by plus (+) or minus (−) signs.
Each term can be a combination of numbers and variables, or just a constant.

Understanding terms is crucial as they form the basis for calculating other polynomial properties, such as the degree.

Degree of a Term

The degree of a term is a key concept in understanding polynomials. It refers to the sum of the exponents of all variables in the term. In simpler terms, it's the power that the variable is raised to. Consider the term $5x^2$ from our earlier example:

The variable $x$ is raised to the power of 2.
Since there is only one variable, the degree of the term is simply 2.

If a term has more than one variable, such as $3x^2y$, the degree would be the sum of the exponents of all variables, which in this case would be $2 + 1 = 3$. Recognizing the degree of each term helps in determining the degree of the entire polynomial.

Exponents in Polynomials

Exponents are integral to understanding polynomials, as they indicate how many times the variable is multiplied by itself. This is often seen in polynomial terms where variables are raised to certain powers. Breaking it down:

The exponent is the small number written above and to the right of the variable.
For $5x^2$, the exponent is 2, meaning $x$ is multiplied by itself once resulting in $x \times x$.
Exponents can also be zero, as in the term $-9$ (the constant term), where the variable part can be seen as $x^0$ which equals 1.

Understanding how exponents work within polynomial terms is pivotal for calculating the degree of terms and the overall degree of the polynomial.

Problem 3

Problem 4

Other exercises in this chapter

Problem 3

For Problems $1-40$, perform the divisions. (Objective 1) $$ \left(x^{2}-5 x-14\right) \div(x-7) $$

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Problem 3

For Problems $1-24$, divide the monomials. $$ \frac{4 x^{3}}{2 x} $$

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Problem 4

For Problems $1-10$, find the indicated products by applying the distributive property; for example, $$ \begin{aligned} (x+1)(y+5) &=x(y)+x(5)+1(y)+1(5) \\ &=

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Problem 4

For Problems $1-30$, multiply using the properties of exponents to help with the manipulation. $$ (9 x)\left(4 x^{3}\right) $$

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