Problem 8

Question

For Problems $1-8$, determine the degree of each polynomial. $$ -7 x+4 $$

Step-by-Step Solution

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Answer

The degree of the polynomial is 1.

1Step 1: Identify Terms in the Polynomial

The given polynomial is $-7x + 4$. Identify each term. Here, $-7x$ and $+4$ are the terms of the polynomial.

2Step 2: Determine the Degree of Each Term

The term $-7x$ has a degree of 1 because the variable $x$ is raised to the power of 1. The term $+4$ has a degree of 0 because it is a constant term without a variable.

3Step 3: Find the Highest Degree

The degree of the polynomial is determined by the term with the highest degree. In this case, $-7x$ has the highest degree, which is 1.

4Step 4: Write the Degree of the Polynomial

The degree of the polynomial $-7x + 4$ is 1, since the highest power of the variable is 1.

Key Concepts

Polynomial TermsDegree of a TermConstant Term in Polynomial

Polynomial Terms

A polynomial is made up of algebraic expressions which can have one or more terms. Each term in a polynomial consists of a coefficient and a variable raised to a power. The expression $-7x + 4$ is a polynomial with two distinct terms.

The first term is $-7x$, which includes the coefficient $-7$ and the variable $x$ raised to the power of 1.
The second term is a constant number, $+4$, which is not attached to any variable.

Understanding the components of each term is crucial for identifying the degree of a polynomial, especially when determining which terms contribute to its degree.

Degree of a Term

The degree of a term is determined by the exponent of the variable, which indicates the term's power. It is an essential concept for identifying the overall degree of a polynomial.

For a term like $-7x$, the degree is 1 because $x$ has an exponent of 1.
When the term is a constant such as $+4$, it is deemed to have a degree of 0, since no variable is present.

To determine the degree of a polynomial, look for the term with the highest degree. In this case, $-7x$ is the term with the highest value of 1.

Constant Term in Polynomial

A constant term in a polynomial, such as $+4$ in the expression $-7x + 4$, is a term without any variable. The degree of this term is always 0. Constant terms can affect the value of the polynomial but not the degree.

They are important because they shift the graph of a polynomial without changing its shape.
In calculations, constants provide fixed values that help define the polynomial's overall behavior, but they do not impact which term determines the polynomial’s degree.

Understanding the role and characteristics of constant terms can clarify aspects of polynomial analysis and aid in solving algebraic expressions involving them.

Problem 8

Problem 9

Other exercises in this chapter

Problem 8

For Problems $1-40$, perform the divisions. (Objective 1) $$ \left(x^{2}-9 x-30\right) \div(x-12) $$

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Problem 8

For Problems $1-24$, divide the monomials. $$ \frac{84 x^{5}}{-7 x^{5}} $$

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Problem 9

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 9

For Problems $1-30$, multiply using the properties of exponents to help with the manipulation. $$ \left(4 a^{2} b^{2}\right)(-12 a b) $$

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