Problem 71 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$\left(2 x^{2}-4 x+1\right)\left(3 x^{2}\right)$$

Step-by-Step Solution

Answer

The product is \(6x^4 - 12x^3 + 3x^2\).

Step 1: Distribute the Monomial

To multiply the polynomial \( (2x^2 - 4x + 1) \) with the monomial \( 3x^2 \), we need to distribute the monomial to each term inside the parentheses. This means multiplying \( 3x^2 \) by each term of the polynomial.

Step 2: Multiply Each Term

Multiply \( 3x^2 \) by \( 2x^2 \): \[3x^2 imes 2x^2 = 6x^4\]Multiply \( 3x^2 \) by \( -4x \):\[3x^2 imes (-4x) = -12x^3\]Multiply \( 3x^2 \) by \( 1 \):\[3x^2 imes 1 = 3x^2\]

Step 3: Combine the Results

Combine the products from Step 2:\[6x^4 - 12x^3 + 3x^2\]

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In the expression \(2x^2 - 4x + 1\), each part is a term. Terms can be simple numbers (constants), variables, or products of numbers and variables. These are essential in algebra as they form the basis of many mathematical operations and equations. In our example, \(2x^2\), \(-4x\), and 1 are different terms of the polynomial. Each term is separated by a plus (+) or minus (-) sign. The power of the variables indicates the degree of each term, with \(2x^2\) having the highest degree. Understanding the structure of algebraic expressions helps us in many operations such as addition, subtraction, and especially multiplication of polynomials. These operations help simplify expressions, solve equations, and analyze relationships between different variables.

Process of Monomial Distribution

Monomial distribution, commonly referred to as the distributive property in algebra, involves dispersing a single term known as a monomial across multiple terms in a polynomial. In the expression \((2x^2 - 4x + 1)(3x^2)\), \(3x^2\) is the monomial.To use monomial distribution:

• Multiply the monomial with each term in the polynomial separately.
• Retain the sign of each term you are distributing to maintain the equation correctly.

In our case, we multiply \(3x^2\) by each of the terms inside the parenthesis one-by-one:

• \(3x^2 \times 2x^2 = 6x^4\)
• \(3x^2 \times -4x = -12x^3\)
• \(3x^2 \times 1 = 3x^2\)

This process ensures that the monomial interacts with every component of the polynomial, maintaining the expression's integrity by systematically managing each term.

Combining Like Terms

Combining like terms simplifies an expression by merging terms that share the same variable component and exponent. However, in our current operation, after distributing the monomial and obtaining \(6x^4 - 12x^3 + 3x^2\), there are no like terms to combine, as each term has a different degree.Understanding when to combine terms:

• Generally, terms like \(x^2\) and \(-3x^2\) can be combined to \(-2x^2\) because they have the same variable at the same power.
• Terms different in degree or variable must remain separate, as they represent different dimensionalities in algebra.

Even though in this problem, combining like terms wasn't necessary, knowing how and when to combine them is crucial for problem-solving efficiency in more complex polynomials. It is a valuable skill for simplifying expressions and making algebraic operations more manageable.