Question

\(h(x)=-5(x+5)(x+1)\)

Short answer

The simplified function is \(h(x)= -10x - 30\).

Step-by-step solution

Identify the given function

The given function is \(h(x)=-5(x+5)(x+1)\). It is a product of several terms.

Apply the distributive property

The distributive property is an algebraic property used to multiply a single term and two or more terms inside a parentheses. Apply it like this: \(-5 \cdot x + -5 \cdot 5\) for the first term and \(-5 \cdot x + -5 \cdot 1\) for the second term, which results in \(-5x -25\) and \(-5x -5\) respectively.

Simplify the function

Combine like terms: \(-5x -5x\) becomes \(-10x\) and \(-25 -5\) becomes \(-30\). Thus, the simplified function becomes \(h(x)= -10x - 30\).

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of one or more variables, usually represented with coefficients. In the exercise given, the function is defined as:

• It's a product of linear factors, meaning it's a form of a polynomial since it includes powers of the variable "x".
• Each term is combined to form a polynomial expression.
• In this case, the polynomial we have is a quadratic polynomial before simplification.

Polynomials can be classified based on their degree:

• The degree is determined by the highest power of the variable within the polynomial expression.
• For example, in the simplified form of the exercise, the degree is 1 because the highest power of "x" is 1.

They are fundamental in different areas of mathematics, such as calculus and algebra, because they can approximate more complex functions.

Simplification

Simplification in mathematics involves reducing an expression to its simplest form. This often means combining like terms and applying mathematical properties to make the expression more straightforward. Let's dive into how this applies to the exercise:

• First, we use the distributive property to expand the product. This gives us individual terms that we can handle easily.
• Next, similar terms are combined. This is when we add or subtract the coefficients of the same variable powers.
• In the exercise, the terms \(-5x - 5x\) were combined to \(-10x\), and \(-25 - 5\) became \(-30\).

The goal of simplification is to make mathematical expressions cleaner and easier to work with, whether it be for solving an equation or understanding its behavior more clearly.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that together form expressions. They're crucial in forming equations and functions, and understanding them can help simplify more complex operations in algebra. In the exercise:

• The expression consists of variables, such as "x," and constants, like the number "5".
• Operations involved include addition and multiplication, crucial in forming the expression.
• Understanding how to manipulate these expressions using algebraic laws like the distributive property is essential—as seen when we expanded \(-5(x+5)\) and \(-5(x+1)\).

These expressions are foundational building blocks in algebra. Making them work through simplification or applying different properties allows us to solve equations and understand their graphical representations or transformations. By mastering algebraic expressions, students can tackle more complex algebraic problems with ease.