Problem 7 - Algebra and Trigonometry

Question

In Exercises 5–8, find the degree of the polynomial. $$ x^{2}-4 x^{3}+9 x-12 x^{4}+63 $$

Step-by-Step Solution

Answer

The degree of the polynomial \(x^{2}-4x^{3}+9x-12x^{4}+63\) is 4.

Step 1 - Arrange the Polynomial

Rearrange the terms of the polynomial in descending order of their powers. Thus the polynomial becomes: \(-12x^{4}+4x^{3}+x^{2}+9x+63\)

Step 2 - Identify the Highest Power

Observe that the highest power of the variable x is 4, which appears in the term \(-12x^{4}\).

Step 3 - Degree of the Polynomial

Since 4 is the highest power of the variable x in the polynomial, the degree of the polynomial is 4.

Key Concepts

Polynomial Order

When we talk about the *order* of a polynomial, we're referring to the degree of the term with the highest power. In simple terms, it is the largest exponent of the variable in the polynomial. For instance, let's consider the polynomial expression: \(-12x^{4} + 4x^{3} + x^{2} + 9x + 63\).
Rewriting the polynomial ensures that the terms are in descending order, starting with the term that has the highest power of the variable. This makes it easier to identify the polynomial order at a glance.

• Descending order helps in systematic analysis and understanding the polynomial's behavior.
• The term that holds the largest exponent is easily spotted at the front, eliminating confusion.

For the example given, the highest power is 4, making the order of the polynomial 4. Always remember, the polynomial's order gives us insight into its overall shape and the kind of terms it has.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants connected by operations like addition, subtraction, multiplication, or division. They do not have an equality sign, differentiating them from algebraic equations.
In the polynomial \(-12x^{4} + 4x^{3} + x^{2} + 9x + 63\), each component like \(-12x^{4}\) or \(9x\), is called a term.

• Terms are the building blocks of algebraic expressions, consisting of a coefficient, a variable, and an exponent.
• Coefficients are the numerical part of the term, affecting its magnitude.
• Constants are the numbers that stand alone without variables, like the \(63\) in our polynomial.

It's vital to recognize these elements to navigate through algebra with ease. Simplifying algebraic expressions involves combining like terms, which have the same variables raised to the same powers.

Highest Power of a Variable

The highest power of a variable in a polynomial is what determines its degree, or order. This highest exponent tells us a lot about the polynomial itself, particularly its degree.
In the polynomial \(-12x^{4} + 4x^{3} + x^{2} + 9x + 63\), the term \(-12x^{4}\) has the highest power of the variable \(x\), which is 4. This power dictates several properties of the polynomial:

• It denotes the degree of the polynomial: here, it's 4.
• The degree can inform us about the potential number of roots and the shape of the graph of the polynomial.
• The leading term (including its sign) influences the end behavior of the polynomial function graph.

Understanding the significance of the highest power of a variable is crucial for analyzing polynomials. It's not just about identifying the degree, but also anticipating how the polynomial behaves as values approach infinity.