Problem 28 - Algebra 1: Concepts and Skills

Question

Write the polynomial in standard form. Then identify the polynomial by degree and by the number of terms. $$ 8+5 y^{2}-3 y $$

Step-by-Step Solution

Answer

The standard form of the given polynomial is \(5y^{2} - 3y + 8\). It is of degree 2 and consists of 3 terms.

Step 1: Reordering the Polynomial into Standard Form

First, reorder the given polynomial, \(8+5 y^{2}-3 y\), into standard form. This involves arranging the polynomial with powers of y in descending order. This provides \(-3y + 5y^{2} + 8\).

Step 2: Put in Proper Order

Putting the terms in proper order results in \(5y^{2} - 3y + 8\)

Step 3: Identifying Degree of the Polynomial

The degree of a polynomial is the highest power of the variable. Looking at \(5y^{2} - 3y + 8\), the highest power is 2. Thus, this polynomial has a degree of 2.

Step 4: Identifying Number of Terms in the Polynomial

The number of terms in this polynomial is 3. These terms are \(5y^{2}\), \(-3y\), and \(8\).

Key Concepts

Standard Form

A polynomial is considered in standard form when its terms are arranged in descending order based on the power of the variable. For instance, if your polynomial involves the variable \(y\), you should order the terms such that the highest power of \(y\) comes first, and the constant comes last.
For the given polynomial \(8 + 5y^2 - 3y\), rearranging the terms in standard form gives us \(5y^2 - 3y + 8\).
Why is this important? Arranging polynomials in standard form makes it easier to read and compare them. It also simplifies operations like addition, subtraction, and division of polynomials, providing clarity and consistency.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial. This is a vital concept because it determines many of the polynomial's properties, including its graph's shape and the number of zeros it may have.
Polynomials can be classified based on their degree:

• Zero Degree: When the highest power of the variable is 0, the polynomial is a constant.
• First Degree: Shaped like a line, frequently called a linear function.
• Second Degree: Forms a parabola, often referred to as a quadratic function.

In our polynomial, \(5y^2 - 3y + 8\), the term \(5y^2\) contains the highest power, which is 2. Therefore, the degree of this polynomial is 2. This makes it a quadratic polynomial.

Number of Terms

The number of terms in a polynomial is straightforward to determine. You simply count the distinct terms present in the expression.
Polynomials can be classified by the number of terms:

• Monomial: A polynomial with just one term (e.g., \(3x^2\)).
• Binomial: A polynomial with two terms (e.g., \(3x + 2\)).
• Trinomial: A polynomial with three terms (e.g., \(x^2 + 2x + 1\)).

Looking at our example polynomial \(5y^2 - 3y + 8\), we observe three distinct terms: \(5y^2\), \(-3y\), and \(8\). Therefore, it is classified as a trinomial.