Problem 110 - Introductory Algebra for College Students

Question

Multiply: \((2 x+3)(x-2)\)

Step-by-Step Solution

Answer

The multiplication of \((2 x+3)(x-2)\) results in \(2x^2 - x - 6\).

Step 1: Apply Distributive Property

Start by multiplying each term of the first binomial with each term of the second. This gives us four terms: \(2x * x\), \(2x * -2\), \(3 * x\), and \(3 * -2\)

Step 2: Simplify the Multiplications

Next, simplify the results of the multiplication to get \(2x^2\), \(-4x\), \(3x\), and \(-6\).

Step 3: Combine Like Terms

Lastly, combine terms having the same variables. In this case, combine \(-4x\) and \(3x\) to get \(-x\), giving us the final simplified expression as \(2x^2 - x - 6\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, applicable in many areas such as multiplying polynomials as seen in this exercise. It tells us how to distribute or spread a factor over terms within parentheses. Essentially, it states that for any three numbers, a, b, and c:

• \(a(b + c) = ab + ac\).

In the case of multiplying two binomials like \((2x + 3)(x - 2)\), apply the distributive property systematically:

• First, multiply each term in the first binomial \((2x + 3)\) by each term in the second binomial \((x - 2)\).
• This leads us to four individual products: \(2x \cdot x\), \(2x \cdot (-2)\), \(3 \cdot x\), and \(3 \cdot (-2)\).

By performing these multiplications, we are using the property to 'distribute' each term of one binomial across each term of the other. This results in simpler expressions that we will later combine.

Binomials

Binomials are algebraic expressions containing two terms, often joined by either a plus or a minus sign. For instance, \(2x + 3\) and \(x - 2\) are examples of binomials. Understanding how to manipulate binomials is crucial in algebra, particularly when multiplying them.

When you multiply two binomials, the goal is to find the product of all possible pairs of terms from each binomial. This process involves a step-by-step distributive approach:

• Each term of the first binomial is multiplied by each term of the second.
• The result is a set of terms that may include both quadratic and linear elements.

Recognizing that dealing with binomials is essentially working with two separate components will help break down more complex polynomial expressions into simpler parts.

Combining Like Terms

Once you have multiplied and obtained all the individual terms from your binomials, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. They can be added or subtracted because they are essentially similar in nature.

• In our example, after distributing and simplifying, you get the terms \(2x^2\), \(-4x\), \(3x\), and \(-6\).
• The like terms in this context are \(-4x\) and \(3x\) as they both contain \(x\) to the same power.

By combining them, you simplify the expression further. Subtract \(4x\) from \(3x\) to get \(-x\). Thus, your expression is now condensed down to \(2x^2 - x - 6\), tidying up the equation into a more manageable form. This process clarifies the expression and reduces errors in more extended calculations.