Problem 61 - Algebra A Combined Function

Question

Multiply. \(-3 x\left(x^{2}+2 x-8\right)\)

Step-by-Step Solution

Answer

The expanded expression is \(-3x^3 - 6x^2 + 24x\).

Step 1: Apply the Distributive Property

To solve this problem, we start by applying the distributive property, which involves multiplying each term inside the parentheses by the term outside. In this case, we multiply \(-3x\) by each term inside the parentheses \(x^2 + 2x - 8\).

Step 2: Multiply \(-3x\) by \(x^2\)

The first term inside the parentheses is \(x^2\). Multiply \(-3x\) by this term:\[-3x \times x^2 = -3x^3\]

Step 3: Multiply \(-3x\) by \(2x\)

Next, multiply \(-3x\) by the second term inside the parentheses, which is \(2x\):\[-3x \times 2x = -6x^2\]

Step 4: Multiply \(-3x\) by \(-8\)

Finally, multiply \(-3x\) by the last term inside the parentheses, which is \(-8\):\[-3x \times -8 = 24x\]

Step 5: Combine the Results

Combine all the results from Steps 2 to 4 to obtain the final expression:\[-3x(x^2) + -3x(2x) + -3x(-8) = -3x^3 - 6x^2 + 24x\]

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, linked together by operations of addition, subtraction, and multiplication. They come with one or more terms, where each term is a product of a constant coefficient and a variable raised to an exponent. For example, in the polynomial expression \(-3x^3 - 6x^2 + 24x\), each term

• \(-3x^3\)
• \(-6x^2\)
• \(24x\)

has its own coefficient and corresponding power of the variable \(x\). Polynomials are frequently used in various branches of mathematics, including algebra and calculus. Understanding how to manipulate them is crucial, as they form the basis of equations and inequalities. In problems involving polynomials, such as the one in our exercise, recognizing each term’s degree (which is the exponent of the variable) helps in organizing and solving the expression effectively.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations such as addition, subtraction, multiplication, and division. Unlike equations, algebraic expressions do not have an equality sign. They often represent real-world quantities and relationships between those quantities. In the given problem, the expression \(-3 x (x^{2} + 2 x - 8)\) is an algebraic expression that we need to simplify by multiplying the terms using the distributive property.
Simplifying algebraic expressions involves combining like terms and performing operations within the expression. The goal is to rewrite it in its simplest form. Once simplified, expressions can be further used for evaluating for specific variable values or in solving equations. As we break the expression down term by term, it becomes easier to manage, especially when dealing with multiple variable terms.

Multiplication of Terms

Multiplication of terms is a fundamental algebraic operation, essential for solving polynomial expressions. In this process, each term from one part of an expression multiplies each term in another part. There are simple steps we follow:

• Multiply the coefficients (the numeric part).
• Multiply the variables by adding their exponents when the base is the same.

For example, when multiplying \(-3x\) by \(x^2\), the coefficients (-3 and 1) are multiplied, and for the variable \(x\), the exponents add up (1 from \(-3x\) and 2 from \(x^2\)) resulting in \(-3x^3\).
Similarly, for the term \(2x\), we multiply the coefficients \(-3\) and \(2\), and again add the exponents for \(x\) to get \(-6x^2\). When multiplying by constants, like \(-8\), only the coefficients multiply, since there are no variable exponents to add, resulting in \(24x\).
Practicing these steps with various expressions enhances skills and helps in understanding later algebraic concepts.