Problem 5 - College Algebra

Question

For the following exercises, identify the degree of the polynomial. $$ 7 x-2 x^{2}+13 $$

Step-by-Step Solution

Answer

The degree of the polynomial is 2.

Step 1: Identifying Terms of the Polynomial

The given polynomial is \(7x - 2x^{2} + 13\). We identify that it consists of three terms: \(7x\), \(-2x^{2}\), and \(13\).

Step 2: Determining the Exponents

For each term, identify the exponent of the variable \(x\). For \(7x\), the exponent is 1 because \(x = x^1\). For \(-2x^{2}\), the exponent is 2. The constant term \(13\) has an exponent of 0.

Step 3: Finding the Highest Exponent

The degree of the polynomial is determined by the highest exponent of \(x\). In this polynomial, the exponents are 1, 2, and 0. Therefore, the highest exponent is 2.

Step 4: Conclusion: Degree of the Polynomial

The polynomial \(7x - 2x^{2} + 13\) has a degree of 2, as the highest power of \(x\) is \(x^{2}\).

Key Concepts

Polynomial Terms

A polynomial is an expression composed of terms.

• Each term consists of a coefficient, a variable raised to an exponent, and sometimes a constant. For example, in the term \( -2x^2 \), -2 is the coefficient, \( x \) is the variable, and 2 is the exponent.
• In our given polynomial \(7x - 2x^2 + 13\), we have three distinct terms: \(7x\), \(-2x^2\), and \(13\).
• Terms are typically separated by a plus \((+)\) or minus \((-))\) sign.

Understanding how to identify terms in a polynomial is crucial as it allows us to determine other properties, such as the degree of the polynomial.

Exponents in Polynomials

Exponents play a critical role in shaping the structure of a polynomial.

• The exponent indicates how many times the variable is multiplied by itself.
• In the expression \(x^n\), \(n\) is the exponent and tells us the 'degree' of that term.
• For instance, in the polynomial \(7x - 2x^2 + 13\), the exponents are 1, 2, and 0, respectively.

The exponents control how steep the curves of the polynomial graph can be. The highest exponent in any of the terms provides the degree of the polynomial.

Constant Term in Polynomials

The constant term in a polynomial is a term that lacks a variable part. It is simply a standalone number.

• In our example, the constant term is \(13\).
• This term can be thought of as \(13x^0\), where the variable \(x\) has an exponent of zero, because any number raised to the power of zero is 1.
• The constant term significantly affects the vertical shift of the polynomial graph on the coordinate plane.

While the constant term does not affect the degree of the polynomial, it is crucial in evaluating the entire expression at specific values of \(x\).