Problem 18 - College Algebra with Modeling and Visualization

Question

Identify the degree and leading coefficient of the polynomial. $$5 x^{2}-x^{3}+7 x^{4}+10$$

Step-by-Step Solution

Answer

The degree is 4 and the leading coefficient is 7.

Step 1: Understand the Polynomial Expression

The given polynomial expression is \(5x^{2} - x^{3} + 7x^{4} + 10\). Note that each term in the polynomial has a coefficient and a power of \(x\) (or is a constant term). Our job is to identify the degree and the leading coefficient of this polynomial.

Step 2: Determine the Degree of the Polynomial

In a polynomial, the degree is the highest exponent (or power) of the variable \(x\). Inspect each term: \(5x^{2}\) has a degree of 2, \(-x^{3}\) has a degree of 3, \(7x^{4}\) has a degree of 4, and the constant term 10 has a degree of 0. Therefore, the highest degree among these is 4.

Step 3: Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. From the previous step, we identified the highest degree to be 4, which corresponds to the term \(7x^{4}\). Therefore, the leading coefficient is 7.

Key Concepts

Degree of a Polynomial

To comprehend the degree of a polynomial, let's think of each term as a little puzzle piece in a big picture. The degree tells us about the highest piece of that puzzle, specifically the tallest tower formed by those puzzle pieces. In our given polynomial expression, we have several terms:

• \(5x^{2}\) with a degree of 2,
• \(-x^{3}\) with a degree of 3,
• \(7x^{4}\) with a degree of 4,
• Constant 10 with a degree of 0.

Among these, the term \(7x^{4}\) holds the highest degree, which is 4. Hence, the degree of this entire polynomial is 4. This degree is crucial because it gives us information about the polynomial's behavior and shape on a graph. For instance, a degree of 4 means the polynomial will potentially have up to 3 turning points.

Leading Coefficient

In a polynomial, every term has a coefficient, and among these, there's a special one called the leading coefficient. This is the coefficient of the term with the highest degree. It's like the leading actor in a movie who stands out among the rest.For our polynomial, we've found out that the highest degree term is \(7x^{4}\). The number "7" here is the coefficient, known as the leading coefficient. Why is this important? Well, it can tell us how 'steep' the graph of the polynomial is. A larger leading coefficient can make the graph stretch higher or lower compared to polynomials with smaller coefficients. Additionally, it can also indicate the growth rate of the polynomial as the variable's values increase or decrease.

Variables and Exponents

Variables and exponents are what make polynomials fascinating. Variables, like \(x\) in our polynomial, are placeholders that can vary, while exponents tell us how many times to multiply that variable by itself.In the expression \(5x^{2} - x^{3} + 7x^{4} + 10\), you see powers of \(x\) ranging from 0 to 4.

• Exponent 2 in \(5x^{2}\) means \(x\times x\),
• Exponent 3 in \(-x^{3}\) means \(x\times x\times x\),
• Exponent 4 in \(7x^{4}\) means \(x\times x\times x\times x\).

Understanding these exponents lets us know how each term contributes to the shape of the polynomial's graph. The higher the exponent, the more complex the term's 'bend' or 'curve'. This makes variables and exponents the key ingredients in the polynomial "soup"—defining its taste, texture, and complexity.