Problem 30 - Algebra 1: Concepts and Skills

Question

Write the polynomial in standard form. Then identify the polynomial by degree and by the number of terms. $$ -2 x+5 x^{3}-6 $$

Step-by-Step Solution

Answer

The polynomial written in standard form is \(5 x^{3} - 2x - 6\). It's a third degree polynomial and it has three terms.

Step 1: Write Polynomial in Standard Form

Polynomial in standard form is written with the terms in order from the highest degree to lowest. The given polynomial is \(-2 x+5 x^{3}-6\). To rewrite this polynomial in standard form, rearrange the terms to go from highest power of x to lowest, which will be \(5 x^{3} - 2x - 6\)

Step 2: Identify the Degree of the Polynomial

Degree of the given polynomial is the highest power of the variable x. Looking at the polynomial \(5 x^{3} - 2x - 6\), the highest power is 3, therefore, the degree of the polynomial is 3.

Step 3: Count the Number of Terms

The number of terms in a polynomial is the number of separated parts by a plus or minus sign. In the given polynomial \(5 x^{3} - 2x - 6\), there are three terms namely: \(5 x^{3}\), \(-2x\), and \(-6\)

Key Concepts

Arranging Polynomials in Standard Form

When we talk about arranging polynomials, we're referring to the process of organizing the terms of a polynomial in a specific order. The standard form of a polynomial requires the terms to be written from the highest degree to the lowest degree. This means you'll start with the term that has the highest exponent and work your way down to the constant term, which has an exponent of zero.

To illustrate, let's consider the polynomial -2x + 5x^3 - 6. When rearranging this polynomial into standard form, we place the terms in descending order of their exponents: the term with x^3 comes first, followed by the x term, and ending with the constant. The standard form would be 5x^3 - 2x - 6. This layout not only makes it easier to read and understand but is also essential for performing operations like addition or subtraction with other polynomials and for polynomial long division.

Why Standard Form Matters

Adopting the standard form is crucial for consistency in mathematical communication. By using a common format, it eliminates confusion and makes it straightforward to compare and perform operations on polynomials. It's a fundamental practice that helps in maintaining clarity when dealing with polynomial expressions.

Degree of a Polynomial

The degree of a polynomial is a fundamental characteristic that informs us about the polynomial's behavior and the shape of its graph. It is determined by the highest power of the variable within the polynomial. To obtain the degree, you don't need to rearrange the polynomial in standard form, though it's typically easier when it is already arranged.

For example, in the polynomial 5x^3 - 2x - 6, the term with the highest exponent is 5x^3, which means the degree of the entire polynomial is 3. The degree can give vital clues about the function, such as the maximum number of roots or the possible number of direction changes in the graph of the polynomial.

Interpreting Polynomial Degree

A higher degree indicates more complexity in the polynomial's graph, like more waves or turns. Knowing the degree helps predict how many times a polynomial will intersect the x-axis, giving a sense of the possible number of zeroes or solutions.

Identifying Polynomial Terms

Polynomial terms are the distinct parts of a polynomial that are added or subtracted. Each term is a product of a constant coefficient and variables raised to whole number exponents. To identify them, look for pieces of the polynomial that are separated by plus or minus signs.

In the standard form polynomial 5x^3 - 2x - 6, there are three terms, which we can list out as follows: 5x^3, -2x, and -6. These terms represent the individual elements that make up the polynomial, with 5x^3 being a cubic term, -2x being a linear term, and -6 being a constant term.

Understanding Each Term

Each term has its role to play in the function's behavior. The leading term, which is the term with the highest degree, largely influences the function's growth. Middle terms can affect the function's slope or rate of change, and the constant term shifts the graph up or down on the coordinate plane.