Problem 69 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$-x\left(2 x^{4}-x^{2}+10\right)$$

Step-by-Step Solution

Answer

The expanded expression is \(-2x^5 + x^3 - 10x\).

Step 1: Identify the terms in the expression

The given expression is \[-x(2x^4 - x^2 + 10)\].This consists of a single term, \(-x\),multiplied by the polynomial \(2x^4 - x^2 + 10\).

Step 2: Distribute the -x across each term of the polynomial

Apply the distributive property, which states \(a(b + c) = ab + ac\), to multiply \(-x\) with each term in the polynomial:1. Multiply \(-x\) with \(2x^4\): \(-x \cdot 2x^4 = -2x^{5}\).2. Multiply \(-x\) with \(-x^2\): \(-x \cdot (-x^2) = x^{3}\).3. Multiply \(-x\) with \(10\): \(-x \cdot 10 = -10x\).

Step 3: Combine the results

After distributing, combine the results of these multiplications to form the expanded expression:\[-2x^5 + x^3 - 10x\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra and plays a vital role in simplifying polynomial expressions. The property states that when you multiply a single term by a sum or difference, you must multiply that single term by each term within the parentheses separately. This can be expressed in the general form:

• For adding: \[ a(b + c) = ab + ac \]
• For subtracting: \[ a(b - c) = ab - ac \]

Let's see how this works. In the exercise \[ -x(2x^4 - x^2 + 10) \], we use the distributive property to multiply \(-x\) with each term inside the parentheses. Following the property ensures that each term gets its due. Hence, \(-x\cdot2x^4\), \(-x\cdot(-x^2)\), and \(-x\cdot10\) are calculated separately, resulting in \(-2x^5\), \(x^3\), and \(-10x\) respectively.
Using the distributive property streamlines complex expressions, making them simpler and easier to work with in further calculations or applications.

Polynomial Expression

Polynomial expressions are key structures in algebra. They're made up of variables, coefficients, and exponents, and they're combined using operations like addition, subtraction, and multiplication. A simple polynomial might look like \(2x^4 - x^2 + 10\).
Polynomials can have various terms:

• Each part of the expression, separated by a plus (+) or minus (-) sign, is called a "term."
• Terms typically consist of a variable raised to an exponent and a constant coefficient.

In our example, \(2x^4\), \(-x^2\), and \(+10\) are the terms. Each one features different variables and exponents. Understanding these components helps in performing algebraic operations, such as the one given in our exercise, where we multiplied the polynomial by \(-x\). This manipulation requires attention to each term, as it shifts and transforms with every operation applied.

Algebraic Operations

In algebra, multiple operations are fundamental for manipulating polynomial expressions. These operations include addition, subtraction, multiplication, and division. Understanding how they affect expressions is crucial for solving algebraic problems effectively.
For polynomial multiplication, like in the exercise \(-x(2x^4 - x^2 + 10)\), you use the distributive property, multiplying each term in the polynomial by \(-x\). Here's how it simplifies the calculation:

• Multiplying powers of the same base adds their exponents. E.g., \(-x \cdot 2x^4 = -2x^{4+1} = -2x^5\).
• Multiply coefficients directly. \(-x\cdot10\) gives \(-10x\).
• For terms involving a double negative, such as \(-x \cdot (-x^2)\), remember that two negatives make a positive, resulting in \(x^3\).

These operations flesh out the polynomial into a simpler form. Mastering them not only helps with specific exercises but also builds a strong foundation for more advanced algebraic processes.