What is a leading coefficient?

In mathematics, specifically in the context of polynomials, the leading coefficient is the coefficient of the term with the highest degree.

• Definition: The leading coefficient is the numerical factor of the term with the highest power of the variable in a <a href="https://www.wikiwhat.page/kavramlar/polynomial">polynomial</a>. For example, in the polynomial 5x^3 + 2x^2 - x + 7, the leading term is 5x^3, and the leading coefficient is 5.

• Significance: The leading coefficient provides essential information about the behavior of a polynomial, especially as the variable approaches positive or negative infinity. The <a href="https://www.wikiwhat.page/kavramlar/leading%20coefficient%20test">leading coefficient test</a> determines the end behavior of the graph of the polynomial.

• Degree of the Polynomial: To identify the leading coefficient, you first need to determine the <a href="https://www.wikiwhat.page/kavramlar/degree%20of%20polynomial">degree of the polynomial</a>, which is the highest power of the variable present.

• Impact on Graph: A positive leading coefficient means that as x approaches positive infinity, the function also tends to positive infinity if the degree of the polynomial is even, and positive infinity if the degree of the polynomial is odd. Conversely, a negative leading coefficient influences the graph in the opposite direction, depending on whether the degree is even or odd.

• Normalization: A polynomial where the leading coefficient is 1 is called a <a href="https://www.wikiwhat.page/kavramlar/monic%20polynomial">monic polynomial</a>.