Problem 75 - Algebra A Combined Function

Question

Multiply. $$ \left(12 x^{3}\right)\left(-x^{5}\right) $$

Step-by-Step Solution

Answer

The product is \(-12x^8\).

Step 1: Recognize the Rule for Multiplying Terms

To multiply two monomials, you multiply their coefficients and then apply the rule that when you multiply like bases, you add their exponents.

Step 2: Multiply the Coefficients

Identify the coefficients from each term. Here, the coefficients are 12 from the term \(12x^3\) and -1 from the term \(-x^5\). Multiply them together: \(12 \times (-1) = -12\).

Step 3: Multiply the Variables

When multiplying variables with the same base, you add their exponents. Here, you multiply \(x^3\) and \(x^5\): Apply the exponent addition rule: \(x^{3+5} = x^8\).

Step 4: Combine the Coefficient and Variable

Combine the results of the coefficient multiplication and the variable multiplication: The product of the monomials is \(-12x^8\).

Key Concepts

Exponent Rules

When working with exponents, it's crucial to remember that they represent repeated multiplication. Understanding the rules that govern exponents makes it easy and clean to simplify expressions. One of the most fundamental rules is the Product of Powers rule. This rule states that when you multiply two exponents with the same base, you add the exponents. For example, if you have \(x^a \times x^b\), you calculate the product by adding the exponents: \(x^{a+b}\).
This rule is used extensively when multiplying monomials, as it simplifies the process of combining similar terms. In the exercise provided, we used this rule to determine that \(x^3 \times x^5 = x^{3+5} = x^8\). Remember, this only works for terms with the same base. Using the same principles can make handling larger expressions or complex problems much easier. It's like a shortcut for dealing with long multiplication of repeated bases.

Coefficient Multiplication

Coefficients in algebra are the numerical factors of the terms, and they play a straightforward role in multiplication. When you multiply monomials, you start by addressing the coefficients. The coefficients are simply numbers, so multiplying them follows the same rules as regular multiplication.

• Find the coefficient of each term.
• Multiply these coefficients.

In the example given, the coefficients are 12 from \(12x^3\) and -1 from \(-x^5\). Therefore, their product is calculated as \(12 \times (-1) = -12\).
The multiplication of coefficients deals directly with basic arithmetic, so practice and familiarity make this step quick and intuitive. Neglecting the signs can lead to incorrect answers, so always remember to consider whether the number you are multiplying is positive or negative. That's how you achieve a correct and simplified coefficient multiplication.

Polynomial Multiplication

Polynomial multiplication involves combining each term in one polynomial with every term in the other polynomial. In the case of monomials, this is simplified down to multiplying a single term by another. This operation involves both the coefficients and the variables.
To multiply monomials, follow these steps:

• Multiply the coefficients to get the new coefficient.
• Use the exponent rules to handle the variables, adding exponents for like bases.
• Combine your results into a single term.

In the exercise \((12x^3) \times (-x^5)\), the steps result in multiplying the coefficients to get \(-12\). Then, applying the exponent rules, you find \(x^8\) from \(x^3 \times x^5\). Combining these results, you get the final product: \(-12x^8\). What makes polynomials easier to manage is identifying like bases, which allows you to organize and simplify even the most complex expressions efficiently. Knowing each component to focus on during the process helps in achieving precise and correct answers.