Problem 58 - Algebra Form and Function

Question

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The degree of \(p(x) q(x)\).

Step-by-Step Solution

Answer

Answer: The degree of the product of the given polynomials p(x) and q(x) is 13.

Step 1: Identify the degrees of the given polynomials

We are given that p(x) is a polynomial of degree 5, and q(x) is a polynomial of degree 8.

Step 2: Recall the property of degrees when multiplying polynomials

When we multiply two polynomials, the degree of their product is the sum of their individual degrees. In this case, we can write: Degree of (p(x) q(x)) = Degree of p(x) + Degree of q(x)

Step 3: Calculate the degree of the product

Using the information given about the degrees of p(x) and q(x), we find the degree of their product: Degree of (p(x) q(x)) = 5 + 8 = 13 So, the degree of the product of the given polynomials, p(x) q(x), is 13.

Key Concepts

Multiplying Polynomials

When we talk about multiplying polynomials, we're referring to the operation where two or more polynomials are combined to form a new polynomial. This is similar to multiplying numbers, but involves variables and exponents. Instead of just multiplying coefficients, we also have to carefully handle the powers of the variable.

Here's a brief guide on how to multiply polynomials:

• Distribute each term in the first polynomial to every term in the second polynomial.
• Multiply the coefficients (the numerical parts) together.
• Combine the exponents of like bases by adding them. For example, multiplying \(x^a\) and \(x^b\) results in \(x^{a+b}\).
• After distributing and simplifying, combine any like terms (terms with the same power of the same variable).

Multiplying polynomials is a fundamental skill in algebra, used to solve equations, simplify expressions, and find polynomial identities.

Degree of Product

The degree of a polynomial is crucial in understanding its behavior and potential roots. When it comes to the degree of a product of polynomials, a useful property comes into play.

The degree of the product of two polynomials is simply the sum of their respective degrees. For example, if you have \(p(x)\) with a degree of 5 and \(q(x)\) with a degree of 8, their product \(p(x)q(x)\) will have a degree of equal to:

• Degree of \(p(x)\) + Degree of \(q(x)\)
• 5 + 8 = 13

This property arises because when you multiply two highest-degree terms from each polynomial, their exponents add up. This sum gives you the degree of the resulting polynomial. Understanding this helps in predicting the complexity of polynomial equations after multiplication.

Polynomial Properties

Polynomials have various intriguing properties that define their structure and behavior. Let's look at some essential properties:

• Degree: The highest exponent in a polynomial indicates its degree, which dictates how the polynomial behaves for large values of its variable.
• Leading Coefficient: This is the coefficient of the term with the highest degree. It influences the shape and direction of the graph of the polynomial function.
• Constant Term: The term without any variables; it is the value of the polynomial when there is no other variable involved, i.e., when the variable is zero.
• Zeroes (or Roots): These are the values of the variable that make the polynomial equal to zero. Identifying the roots helps solve polynomial equations.

Understanding these basics not only aids in algebraic manipulations but also prepares students for more advanced topics like calculus and more complex algebraic expressions. The consistent practice of recognizing and utilizing these properties in polynomial operations is vital in mastering algebra.