Problem 67 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Distribute the Monomial

To multiply the monomial \( 3x \) with the polynomial \( (2x^2 - x - 1) \), we will distribute \( 3x \) to each term inside the parenthesis. This means multiplying \( 3x \) by each term one at a time.

Step 2: Multiply Each Term

First, multiply \( 3x \) by \( 2x^2 \): \ \[ 3x imes 2x^2 = 6x^3 \] \ Next, multiply \( 3x \) by \( -x \): \ \[ 3x imes (-x) = -3x^2 \] \ Finally, multiply \( 3x \) by \( -1 \): \ \[ 3x imes (-1) = -3x \]

Step 3: Combine the Terms

Write down all the terms obtained from the previous calculations: \ \[ 6x^3 - 3x^2 - 3x \] \ Since there are no like terms to combine, this expression is already simplified.

Key Concepts

Monomial Distribution

When we perform monomial distribution, we are essentially spreading one single term, the monomial, across all the terms in a polynomial. This concept is essential in polynomial multiplication as it breaks down the multiplication process into smaller, simpler steps.

In the original exercise, the problem requires distributing the monomial \(3x\) onto each term of the expression \((2x^2 - x - 1)\). Each term in the polynomial is successively multiplied by the monomial.
For instance:

• The first distribution involves \(3x\) and \(2x^2\), resulting in \(6x^3\).
• The second distribution is \(3x\) and \(-x\), leading to \(-3x^2\).
• The last distribution combines \(3x\) with \(-1\), giving \(-3x\).

This method ensures that each term in the polynomial is handled individually, helping to avoid mistakes and making calculations clearer.

Combining Like Terms

Combining like terms is a crucial step in simplifying polynomial expressions after carrying out operations like addition, subtraction, or multiplication. Like terms are those terms in the polynomial that have the exact same variable parts raised to the same power.

In the original problem's solution, after expanding through monomial distribution, you end up with the expression \(6x^3 - 3x^2 - 3x\). Here, there are no like terms to combine:

• \(6x^3\) stands alone due to its different power level compared to the others.
• \(-3x^2\) and \(-3x\) each also differ from one another.

Therefore, the expression is already fully simplified. Always look for like terms after expanding to simplify expressions when possible.

Polynomial Expressions

Polynomial expressions are made up of terms that are added or subtracted together. Each term consists of a coefficient and one or more variables raised to a power. In the context of multiplying polynomials, the goal is to use mathematical operations to simplify or expand the expressions.

The polynomial \((2x^2 - x - 1)\) has three distinct terms:

• \(2x^2\), which is a quadratic term,
• \(-x\), which is a linear term,
• and \(-1\), a constant term.

Multiplying polynomials involves paired combinations of these terms with the distributed monomial. Understanding the behavior of different powers when multiplied is central to mastering polynomials. For example, multiplying two terms with variables results in summing their exponents: \(x^2\) multiplied by \(x\) gives \(x^3\), thus expanding the polynomial expression into a new form.