Problem 45 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \left(2 x^{3}\right)\left(-4 x^{2}\right) $$

Step-by-Step Solution

Answer

The result is \(-8x^{5}\).

Step 1: Identify the Multiplication Expression

We have the expression \((2 x^{3})(-4 x^{2})\), which involves multiplying coefficients and variables with exponents.

Step 2: Multiply the Coefficients

Multiply the numerical coefficients from each term: \(2\) and \(-4\):\[2 \times (-4) = -8.\]

Step 3: Multiply the Variables with Exponents

Use the rule of exponents to multiply the variables. The rule \(x^a \times x^b = x^{a+b}\) applies here.\[x^{3} \times x^{2} = x^{3+2} = x^{5}.\]

Step 4: Combine the Results

Combine the result of the coefficients and the result of the variables to get the final expression:\[-8x^{5}.\]

Key Concepts

Coefficients

In polynomial multiplication, coefficients play a crucial role as they determine the magnitude of the terms involved. When multiplying expressions, like \((2x^3)\) and \((-4x^2)\), the coefficients are the numbers that precede the variables. Here, they are 2 and -4.

To find the result of the multiplication, you begin by multiplying these coefficients together. The calculation is simple: you multiply 2 by -4, which results in -8.

There are a few important things to remember about coefficients:

• Coefficients can be any real number, including zero or negative numbers.
• Multiplying coefficients follows the normal arithmetic rules for multiplying numbers.

Accurately handling coefficients is key to obtaining the correct numerical part of your answer in polynomial multiplication.

Exponents

Exponents are the numbers that indicate how many times a variable is multiplied by itself. In our original expression, \((2x^3)\) and \((-4x^2)\), the exponents are 3 and 2, respectively, attached to the variable \(x\).

When multiplying variables with exponents, use the product of powers property, which involves adding the exponents together, rather than multiplying them. So for our terms, \(x^3\) and \(x^2\), you add 3 and 2 together to get 5. This means, \(x^3 \times x^2 = x^{3+2} = x^5\).

Key points about working with exponents include:

• Always ensure that you're dealing with the same base before applying the product of powers rule.
• Add the exponents when multiplying, but remember to subtract when dividing.

Understanding how to manipulate exponents is vital in simplifying expressions to their most basic form.

Product of Powers

The product of powers is an essential rule for simplifying expressions with exponents. This rule states that when you multiply two powers that have the same base, you can add the exponents.

In the multiplication \(x^3 \times x^2\), the base \(x\) is the same, so you add the exponents: 3 and 2, resulting in \(x^5\). This simple addition is the core of applying the product of powers rule.

To effectively apply this concept, keep these guidelines in mind:

• This property only applies to like bases (e.g., \(x\) with \(x\), \(y\) with \(y\)).
• Double-check that your bases are consistent before adding exponents.
• Be cautious with signs; though exponents are added, coefficients are multiplied normally, which can affect the sign of the outcome.

By mastering the product of powers, you can easily tackle and simplify more complex polynomial multiplications and algebraic expressions.