In Exercises 5–8, find the degree of the polynomial. 3x²−5x+4

Verified step by step guidance

1. Identify the terms of the polynomial: The given polynomial is 3x^2 - 5x + 4. The terms are 3x^2, -5x, and 4.
2. Determine the degree of each term: The degree of a term is the exponent of the variable in that term. For 3x^2, the degree is 2; for -5x, the degree is 1; and for 4 (a constant), the degree is 0.
3. Find the highest degree among the terms: Compare the degrees of all terms. The degrees are 2, 1, and 0. The highest degree is 2.
4. Conclude that the degree of the polynomial is the highest degree of its terms: Since the highest degree is 2, the degree of the polynomial is 2.
5. Verify your understanding: The degree of a polynomial is determined solely by the term with the highest exponent of the variable, regardless of the coefficients or constant terms.

Key Concepts

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the polynomial's behavior and the number of roots it can have. For example, in the polynomial 3x^2−5x+4, the highest exponent is 2, making the degree of this polynomial 2.

Polynomial Structure

A polynomial is an algebraic expression that consists of terms, each of which is a product of a constant coefficient and a variable raised to a non-negative integer exponent. The general form of a polynomial in one variable is a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n are coefficients and n is a non-negative integer.

Coefficients

Coefficients are the numerical factors in a polynomial term. In the polynomial 3x^2−5x+4, the coefficients are 3 for x^2, -5 for x, and 4 as the constant term. Understanding coefficients is essential for analyzing the polynomial's properties, such as its shape and intercepts.