Problem 12 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((3 x+1)(x-2)\)

Step-by-Step Solution

Answer

The simplest form of the expression is \(3x^2 - 5x - 2\).

Step 1: Apply the Distributive Property

To simplify the expression \((3x+1)(x-2),\) apply the distributive property (also known as the FOIL method for binomials). This means multiplying each term in the first binomial by each term in the second binomial.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \((3x)\) and \(x\).\[3x \cdot x = 3x^2\]

Step 3: Multiply the Outer Terms

Multiply the outer terms in the binomials: \((3x)\) and \(-2\).\[3x \cdot (-2) = -6x\]

Step 4: Multiply the Inner Terms

Multiply the inner terms in the binomials: \((1)\) and \(x\).\[1 \cdot x = x\]

Step 5: Multiply the Last Terms

Multiply the last terms in the binomials: \((1)\) and \(-2\).\[1 \cdot (-2) = -2\]

Step 6: Combine Like Terms

Add up all the products:\[3x^2 - 6x + x - 2\]Combine like terms:\[3x^2 - 5x - 2\]

Key Concepts

Understanding Binomials

A binomial is a type of algebraic expression that has exactly two terms. It's like having just two items in your shopping list and calling it a list. In mathematics, a term is composed of numbers or variables, often linked by multiplication or division.
For example, in the expression \((3x+1)\), there are two terms:

• \(3x\)
• \(1\)

Similarly, in the expression \((x - 2)\), the two terms are:

• \(x\)
• \(-2\)

When dealing with expressions like \((3x+1)(x-2)\), the focus is on finding the product of these binomials. Each term in the first binomial will be multiplied with each term in the second, which leads us to the next concept.

Breaking Down the FOIL Method

The FOIL method is a straightforward way to multiply two binomials. FOIL stands for First, Outer, Inner, and Last, which are the steps to follow when multiplying. It's like a map guiding you through the binomial multiplication process.

• First: Multiply the first terms of each binomial. This step helps to form the starting part of your final expression. For example, multiplying \(3x\) with \(x\) gives \(3x^2\).
• Outer: Multiply the outermost terms of the binomials. With \(3x\) and \(-2\), you arrive at \(-6x\).
• Inner: Multiply the inner terms, which are closest to the center of the expression. Multiplying \(1\) by \(x\) results in \(x\).
• Last: Multiply the last terms of each binomial to complete the expression. Here, \(1\) times \(-2\) equals \(-2\).

This technique neatly organizes the process, making it easier to ensure no terms are missed.

Simplifying Expressions After FOIL

Once all the products from the FOIL process have been listed out, the next important step is simplifying the expression. Simplification involves combining any like terms.
Often, the result from FOIL yields what seems to be cluttered algebra. In our example, the expression after using FOIL was:
\[3x^2 - 6x + x - 2\]
By identifying and combining like terms, it becomes more manageable. Here, \(-6x\) and \(x\) can be combined because they both have the \(x\) variable:

• Combine: \(-6x\) plus \(x\) gives us a simplified \(-5x\)

After combining, the expression becomes:
\[3x^2 - 5x - 2\]. After simplifying, you have a clear and concise expression.

Identifying and Combining Like Terms

Like terms are terms that have the same variable raised to the same power, even if their coefficients are different. Recognizing them is crucial for simplifying expressions effectively.
For example, in \(3x^2 - 6x + x - 2\), both \(-6x\) and \(x\) are like terms because they both contain the same variable \(x\) to the same power (the first power).
When you combine these, think of it as gathering similar tasks. It's about grouping together things that are alike.

• The coefficients of these terms, \(-6\) and \(+1\), merge to give \(-5x\).

This step reduces the complexity of the expression, making it simpler to solve or further manipulate. Always check for like terms as a part of the simplification process.