Problem 22 - Algebra Form and Function

Question

Give the leading coefficient. $$ 100-\sqrt{6} s+15 s^{2} $$

Step-by-Step Solution

Answer

Answer: The leading coefficient is \(15\).

Step 1: Identify the highest degree term

The polynomial expression is given as: $$100-\sqrt{6}s + 15s^2.$$ The highest degree term is the one with the exponent associated with the variable "s", which is \(15s^2.\)

Step 2: Locate the coefficient

In the term \(15s^2\), the coefficient is the number multiplying the variable \(s^2\). In this case, the coefficient is \(15\).

Step 3: Conclusion

The leading coefficient of the given polynomial expression $$100-\sqrt{6}s + 15s^2$$ is \(15\).

Key Concepts

Polynomial Expression

A polynomial expression is a mathematical phrase involving a sum of powers of one or more variables multiplied by coefficients. Each part of the polynomial, separated by plus or minus signs, is called a term. Specifically:

• The power, or exponent, tells you how many times to multiply the variable by itself.
• The coefficient is the number in front of the variable that indicates how many of each term there are.

For instance, in the expression \(100 - \sqrt{6}s + 15s^2\), there are three terms: 100, \(-\sqrt{6}s\), and \(15s^2\). Each term can have a coefficient and a power associated with the variable.

Highest Degree Term

Understanding the highest degree term in a polynomial is crucial because it determines many characteristics of the polynomial, including its leading coefficient and its behavior as the variable becomes very large or small.

• The degree of a term is defined by the exponent of the variable in that term.
• The term with the highest degree is called the leading term.

In the polynomial \(100 - \sqrt{6}s + 15s^2\), we look for the term where the exponent of \(s\) is the largest. Here, the term \(15s^2\) has the highest degree, which is 2, because the exponent on \(s\) is two. This term is what we evaluate first to understand the polynomial's behavior as well as to identify its leading coefficient.

Coefficient

A coefficient in a polynomial is the number that multiplies a term with a variable in it. Coefficients can be integers, fractions, decimals, or irrational numbers.

• The leading coefficient is found in the term with the highest degree.
• They are important for determining the shape and orientation of a polynomial graph.

In the expression \(100 - \sqrt{6}s + 15s^2\), the coefficient of the term with the highest degree \(s^2\) is 15. This makes 15 the leading coefficient. Coefficients tell us how steep or flat the graph of the polynomial will be and contribute to the direction it will take as the variable values change.