Question

$$ \text { perform the indicated operations and simplify. } $$ $$ (3 x-9)(2 x+1) $$

Short answer

The simplified expression is \(6x^2 - 15x - 9\).

Step-by-step solution

Distribute Each Term

To solve the multiplication, we will distribute each term in the first binomial to each term in the second binomial. For the binomials \((3x - 9)(2x + 1)\), this means: 1. Multiply \(3x\) by \(2x\) to get \(6x^2\). 2. Multiply \(3x\) by \(1\) to get \(3x\). 3. Multiply \(-9\) by \(2x\) to get \(-18x\). 4. Multiply \(-9\) by \(1\) to get \(-9\). Our expression is now \(6x^2 + 3x - 18x - 9\).

Combine Like Terms

Next, we combine the like terms from the distributed expression. The like terms here are \(3x\) and \(-18x\). Adding them results in:\(6x^2 + (3x - 18x) - 9\), which simplifies to:\(6x^2 - 15x - 9\).

Final Simplified Expression

The expression \(6x^2 - 15x - 9\) is already in its simplest form, as no further like terms are present. Thus, this will be the final simplified answer for the problem.

Key concepts

Binomial Multiplication

Binomial multiplication involves expanding a product of two binomials, such as \((3x - 9)(2x + 1)\). The goal here is to perform multiplication across all terms from both binomials. This operation results in an expression with multiple terms that need to be simplified.

To effectively multiply two binomials:

• Identify each term in the first binomial.
• Multiply each term in the first binomial by each term in the second binomial.
• This process will yield four individual products that need to be arranged into a full expression.

In this exercise, we start with terms \((3x - 9)\) and \((2x + 1)\). Multiplying each of these terms as outlined creates the components \(6x^2 + 3x - 18x - 9\). The next step is to simplify this by combining like terms.

Distribution

Distribution is a crucial step in the process of multiplying binomials. It involves spreading each term of the first polynomial or binomial over every term of the second one.

In the example provided, we systematically distribute as follows:

• First, multiply \(3x\) from the first binomial by each term in the second binomial: \(3x \times 2x = 6x^2\) and \(3x \times 1 = 3x\).
• Next, distribute \(-9\) to the second binomial's terms: \(-9 \times 2x = -18x\) and \(-9 \times 1 = -9\).

The results of these distributions combine to form the expression: \(6x^2 + 3x - 18x - 9\).

By carefully carrying out this operation, we ensure all parts of each binomial are multiplied appropriately, covering all diagonal and horizontal interactions within the multiplication.

Like Terms

When simplifying expressions, the identification and combining of like terms is essential. Like terms are those that have identical variable parts raised to the same power.

In the expression \(6x^2 + 3x - 18x - 9\), the like terms are \(3x\) and \(-18x\) since both entail the linear term \(x\).

To simplify, add or subtract these coefficients:

• \(3x - 18x = -15x\).

The result simplifies the expression to \(6x^2 - 15x - 9\).

Remember, constant or different power terms like \(6x^2\) or \(-9\) are not combined with others unless they match entirely in type and power. Recognizing and combing like terms streamlines your polynomial expressions, reducing them to simpler forms that are easier to work with.

Polynomial Simplification

Simplifying polynomials means reorganizing the expression to its most reduced form by eliminating redundancies like multiple same-term expressions and combining all like terms.

In this case, we started with \(6x^2 + 3x - 18x - 9\). Upon identifying and determining like terms, we combine \(3x - 18x\) into \(-15x\).

What remains is a streamlined expression: \(6x^2 - 15x - 9\). There are no further simplifications possible because there are no more like terms or fractional constants.

Tips for simplification include:

• Group and simplify like terms first.
• Maintain the order by arranging terms from highest to lowest degree.
• Look out for possible factorizations if required by further steps.

Always ending with a completely simplified polynomial ensures clarity and correctness in algebraic operations.