Problem 7 - Precalculus Mathematics for Calculus

Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial; then list its terms and state its degree. $$x^{2}-3 x+7$$

Step-by-Step Solution

Answer

Trinomial; terms: \(x^2, -3x, 7\); degree: 2.

Step 1: Identify the Type of Polynomial

Examine the expression \(x^2 - 3x + 7\). Count the number of terms to determine the type of polynomial. It has three terms \(x^2\), \(-3x\), and \(7\), so it is a trinomial.

Step 2: List the Terms

List each term of the polynomial separately: the terms of \(x^2 - 3x + 7\) are \(x^2\), \(-3x\), and \(7\).

Step 3: Determine the Degree

To find the degree, identify the highest power of the variable in the polynomial. Here, the highest power is \(2\), from the term \(x^2\). Thus, the degree of the polynomial is \(2\).

Key Concepts

Monomial

A monomial is the simplest form of a polynomial. Think of it as a single piece of the algebra puzzle. It consists of just one term. This term can be a number, a variable, or a combination of numbers and variables multiplied together. Here are some key aspects of monomials:

• Contains only one term, such as $3x^2$, $-7x$, or $10$.
• The term can include a coefficient (a constant number), a variable (such as $x$), and an exponent (a number that tells how many times to multiply the variable by itself).
• When identifying a monomial, remember that there are no addition or subtraction operations separating it into multiple terms.

Monomials are the building blocks for larger polynomials and help establish more complex expressions.

Binomial

When you hear binomial, think of two. A binomial is a type of polynomial that contains exactly two terms, joined by a plus or minus sign. This structure makes binomials a bit more versatile than monomials:

• Consists of two terms, like $x + 5$ or $3x^2 - 4$.
• Each term in a binomial can be a monomial, and the terms can have different powers or coefficients.
• Important in algebraic operations, binomials serve as a basis for polynomial multiplication, factoring, and expansion exercises using the binomial theorem.

Understanding binomials can simplify tackling problems involving expressions with two terms.

Trinomial

A trinomial, as its name hints, involves three terms. Just like monomials and binomials, trinomials are polynomials, but with a trio of terms that are added or subtracted. Here's what you need to know:

• Comprised of three terms, such as $x^2 - 3x + 7$ or $2a^2 - a + 1$.
• Each term can vary in degree, coefficient, or variable, adding to the complexity of the expression.
• Trinomials provide a foundation for exploring algebraic concepts like factoring into simpler polynomials or applying the quadratic formula.

Grasping the idea of trinomials is key for more advanced algebraic manipulations and solutions.

Degree of a Polynomial

The degree of a polynomial is a way to express the complexity or the highest level of its terms. It tells us the maximum number of times a variable in the polynomial is raised to a power. Understanding the degree of a polynomial helps simplify evaluation and comparison:

• Determined by identifying the term with the highest exponent, like the $x^2$ in $x^2 - 3x + 7$.
• If a polynomial, say $3x^3 - 2x + 4$, has degrees of $3$ for $x^3$, $1$ for $x$, and a constant with degree $0$, the degree is $3$.
• The degree can impact polynomial functions' behavior, like determining long-term trends or influencing the graph's shape.

In practice, recognizing the degree aids in predicting the behavior of polynomials in various mathematical and real-world applications.