In Exercises 5–8, find the degree of the polynomial. x²−4x³+9x−12x⁴+63

Verified step by step guidance

1. Step 1: Understand the concept of the degree of a polynomial. The degree of a polynomial is the highest power of the variable (in this case, x) with a non-zero coefficient.
2. Step 2: Identify all the terms in the polynomial. The given polynomial is: $x^2\; -\; 4x^3\; +\; 9x\; -\; 12x^4\; +\; 63$. The terms are: $x^2$, $-4x^3$, $9x$, $-12x^4$, and $63$.
3. Step 3: Determine the degree of each term. The degree of a term is the exponent of the variable x. For the terms: $x^2$ has degree 2, $-4x^3$ has degree 3, $9x$ has degree 1, $-12x^4$ has degree 4, and $63$ (a constant) has degree 0.
4. Step 4: Identify the highest degree among all the terms. From the degrees calculated in Step 3, the highest degree is 4, which comes from the term $-12x^4$.
5. Step 5: Conclude that the degree of the polynomial is 4, as it is the highest power of x with a non-zero coefficient.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the polynomial's behavior as the variable approaches infinity and helps determine the number of roots and the shape of its graph. For example, in the polynomial x^2 - 4x^3 + 9x - 12x^4 + 63, the degree is determined by the term with the highest exponent.

Polynomial Terms

A polynomial is composed of terms, which are individual components that can include constants, variables, and exponents. Each term is typically in the form of ax^n, where 'a' is a coefficient, 'x' is the variable, and 'n' is a non-negative integer. Understanding how to identify and classify these terms is essential for determining the degree of the polynomial.

Standard Form of a Polynomial

The standard form of a polynomial arranges its terms in descending order of their exponents. This format makes it easier to identify the leading term and the degree of the polynomial. For instance, rewriting the polynomial x^2 - 4x^3 + 9x - 12x^4 + 63 in standard form would help quickly identify the term with the highest degree, which is crucial for solving the problem.