Problem 3 - Algebra Form and Function

Question

Are the functions power functions? $$ y=3 x^{3}+2 x^{2} $$

Step-by-Step Solution

Answer

Answer: No, the given function is not a power function since it contains two terms with different exponents and cannot be simplified into a single-term power function.

Step 1: Write down the given function expression

The given function is: $$ y=3x^3+2x^2 $$

Step 2: Check if the function is in the form of a power function

In a power function, there should be only one term, and it should be in the form \(ax^p\). However, the given function has two terms and cannot be simplified into a single term: $$ y = 3x^3 + 2x^2 \neq ax^p $$

Step 3: Conclusion

Since the given function has two terms with different exponents and cannot be simplified into a single term, it does not fulfill the requirement of a power function. Therefore, the given function is not a power function.

Key Concepts

Polynomial Functions

Polynomial functions are fascinating algebraic expressions that consist of one or more terms. Each term is made up of a coefficient (a number) and a variable raised to an exponent. For instance, in the function \( y = 3x^3 + 2x^2 \), there are two distinct terms: \( 3x^3 \) and \( 2x^2 \). Each has a coefficient (3 and 2, respectively) and a variable \( x \) raised to a power (3 and 2).Polynomials can be classified by the highest power of the variable in any single term, known as the degree. In our example, the highest exponent is 3, so it's a polynomial of degree 3. Further down the line, the degree tells you important things about the polynomial, like the general shape of its graph.Understanding how to identify and work with polynomial functions helps in solving equations, analyzing graphs, and more. They are like the building blocks for many functions you will encounter in algebra and calculus.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division). The beauty of algebraic expressions is their ability to compactly represent patterns in numbers, which is essential for problem-solving.In the function \( y = 3x^3 + 2x^2 \), this expression combines two algebraic terms \( 3x^3 \) and \( 2x^2 \). Each is an expression in itself, comprising a coefficient, a variable, and an exponent.Recognizing different parts of an algebraic expression is crucial:

• Coefficient: The number in front of the variable (e.g., 3 and 2 in the example).
• Variable: The letter that represents a number (\( x \) in the example).
• Exponent: The power to which the variable is raised (3 and 2 in the example).

This understanding aids in simplifying expressions, solving algebraic equations, and more.

Exponent Rules

Exponents are shorthand for repeated multiplication of a number by itself. Exponent rules help in simplifying algebraic expressions, such as the ones found in polynomial functions.Let's explore some key exponent rules that can help in dealing with expressions like \( y = 3x^3 + 2x^2 \):

• Product Rule: To multiply two expressions with the same base, you add the exponents: \( x^a \, x^b = x^{a+b} \).
• Power Rule: To raise an expression to an exponent, you multiply the exponents: \( (x^a)^b = x^{a \, b} \).
• Zero Exponent Rule: Any base raised to the zero power is 1: \( x^0 = 1 \).
• Negative Exponent Rule: A negative exponent means the reciprocal of the base raised to the positive exponent: \( x^{-a} = \frac{1}{x^a} \).

These rules are like the grammar of algebra, they determine how expressions are manipulated and simplified. Understanding and applying them makes working with algebraic expressions more manageable and significantly expands the toolkit for solving mathematical problems.