Problem 17 - Algebra 2

Question

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(7+3 x^{2}-5 x^{3}+6 x^{2}-2 x\)

Step-by-Step Solution

Answer

Degree: 3, Leading Coefficient: -5.

Step 1: Combine Like Terms

First, we need to combine like terms in the given polynomial expression. The expression is: \(7 + 3x^{2} - 5x^{3} + 6x^{2} - 2x\). Combine the terms with \(x^2\): \(3x^{2} + 6x^{2} = 9x^{2}\). The simplified expression becomes \(-5x^{3} + 9x^{2} - 2x + 7\).

Step 2: Identify Degree of the Polynomial

Now, identify the degree of the polynomial. The degree of a polynomial is the highest power of the variable \(x\) in the expression. In the simplified form \(-5x^{3} + 9x^{2} - 2x + 7\), the term with the highest power is \(-5x^{3}\), so the degree of this polynomial is 3.

Step 3: Determine the Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. We've identified the term with the highest degree as \(-5x^{3}\). Therefore, the leading coefficient is \(-5\).

Step 4: Verify It Is a Polynomial in One Variable

Check that the expression is a polynomial in one variable. The given expression only contains powers of the same variable \(x\), with each term having a constant coefficient. Thus, it is a polynomial in one variable.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is crucial in determining its characteristics and behavior. The degree refers to the highest power of the variable present in the polynomial expression. For example, consider the polynomial expression \[-5x^{3} + 9x^{2} - 2x + 7\].Here, the degree is determined by the term with the highest exponent of the variable, which is \(-5x^{3}\).Thus, this polynomial's degree is 3. Identifying the degree provides valuable information about the polynomial, such as its general form and the number of turning points or intersections with the x-axis that its graph might have.
The degree also affects the end behavior of the polynomial, giving insights into how the polynomial behaves as the variable takes on very large positive or negative values.

Leading Coefficient

The leading coefficient of a polynomial is another important feature that often surfaces in polynomial analysis. It refers specifically to the coefficient of the term that has the highest degree.In the polynomial \(-5x^{3} + 9x^{2} - 2x + 7\),the highest degree term is \(-5x^{3}\),making \(-5\)the leading coefficient.This coefficient plays a significant role in influencing the shape of the polynomial's graph. When examining whether a graph opens upwards or downwards, or its steepness, the leading coefficient is the key player.
Additionally, the sign of the leading coefficient can indicate whether the ends of the graph are rising or falling as it extends toward infinity.

Polynomial in One Variable

To understand a polynomial in one variable, imagine a mathematical expression constructed from constants and a single variable raised to various powers. For the polynomial in question, the expression is\[-5x^{3} + 9x^{2} - 2x + 7\].This expression is considered a polynomial in one variable. It uses the variable \(x\)throughout and maintains constant coefficients for each term. Key characteristics include:

• Uniqueness of the variable: Only one variable, \(x\),is involved.
• Combination of various power terms: Powers may range, such as the terms with degrees 0 (constant term), 1, 2, and 3.
• Non-negative integer exponents: The exponents of \(x\)are all non-negative integers.

A polynomial in one variable forms a fundamental building block in algebra, making equations easier to solve and analyze.
This type of polynomial is widely used in modeling real-world phenomena and solving equations, thanks to its relatively straightforward nature.