Problem 70 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$-2 x^{2}\left(5 x^{3}+x^{2}-2\right)$$

Step-by-Step Solution

Answer

The result is \(-10x^5 - 2x^4 + 4x^2\).

Step 1: Distribute the First Term

To multiply the polynomial, start by distributing the term \(-2x^2\) to each term inside the parentheses: \(-2x^2 (5x^3 + x^2 - 2)\).

Step 2: Multiply Each Term

Multiply \(-2x^2\) by each term within the parentheses, keeping in mind the laws of exponents and multiplication:- First, calculate: \(-2x^2 \cdot 5x^3 = -10x^{5}\)- Next, calculate: \(-2x^2 \cdot x^2 = -2x^{4}\)- Finally, calculate: \(-2x^2 \cdot (-2) = 4x^{2}\)

Step 3: Write the Final Expression

Combine the multiplied terms to write the final expression:\(-10x^5 - 2x^4 + 4x^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a polynomial expression. This process makes it possible to handle complex expressions systematically, breaking them down into simpler calculations. In the exercise, the distributive property is used to multiply the term \(-2x^2\) by each element within the parentheses \((5x^3 + x^2 - 2)\). By doing so, we separate the operation into manageable parts.

- First, multiply \(-2x^2\) by \(5x^3\), yielding \(-10x^5\).
- Next, \(-2x^2\) multiplies \(x^2\), resulting in \(-2x^4\).
- Finally, \(-2x^2\) is multiplied by \(-2\), giving \(4x^2\).

This property effectively distributes the multiplication across all terms, facilitating straightforward simplification of polynomial expressions.

Laws of Exponents

The laws of exponents are essential for simplifying expressions involving powers of the same base. When multiplying exponents, it is vital to apply these laws correctly. The rule used in our exercise is that when you multiply two terms with the same base, you add the exponents.
Sometimes this concept is phrased as \(a^m \cdot a^n = a^{m+n}\).

For example, when you multiply \(-2x^2\) by \(5x^3\), you're essentially performing:
- Multiply the coefficients (-2 and 5), resulting in -10.
- Add the exponents of the base \(x\) (2 + 3), getting \(x^5\).
So, \(-2x^2 \cdot 5x^3 = -10x^5\). This approach also applies to other terms in the exercise, such as \(-2x^2 \cdot x^2\), where the exponents of \(x\) are added (2 + 2) to get \(-2x^4\).

Remember, understanding and applying these laws correctly is key to simplifying polynomial expressions efficiently.

Polynomial Expressions

Polynomial expressions are mathematical phrases that consist of variables raised to natural number exponents and have coefficients. These expressions can vary in complexity, ranging from a single term to several terms combined through addition or subtraction.
When working with polynomials, we often perform operations like addition, subtraction, and, as in this exercise, multiplication.

• Consider a polynomial like \(5x^3 + x^2 - 2\). It includes different terms with varying degrees (where the degree is the highest power of the variable within the polynomial).
• In multiplying polynomials, as shown in the original exercise, each term within the first expression distributes to every term in the other expression, leading to a new polynomial composed of the summed products.

The final expression obtained \(-10x^5 - 2x^4 + 4x^2\) after performing polynomial multiplication, is also a polynomial, with ordered terms according to their degree. Understanding these elements will strengthen your abilities to solve higher math problems involving polynomials.