Problem 57 - College Algebra with Modeling and Visualization

Question

Multiply the binomials. $$(3-2 x)(3+x)$$

Step-by-Step Solution

Answer

The product is \(-2x^2 - 3x + 9\).

Step 1: Apply the Distributive Property

The first step in multiplying the binomials is to apply the distributive property, also known as the FOIL method (First, Outside, Inside, Last). Essentially, we multiply each term in the first binomial by each term in the second binomial. Start by multiplying the first terms from each binomial: \[(3) imes (3) = 9\]

Step 2: Multiply the Outside Terms

Next, multiply the outside terms from each binomial together: \[(3) imes (x) = 3x\]

Step 3: Multiply the Inside Terms

Now, multiply the inside terms from each binomial: \[(-2x) imes (3) = -6x\]

Step 4: Multiply the Last Terms

Finally, multiply the last terms from each binomial: \[(-2x) imes (x) = -2x^2\]

Step 5: Combine Like Terms

Now, add up all the terms obtained from each multiplication: \[9 + 3x - 6x - 2x^2\]Combine the like terms \(3x\) and \(-6x\), which results in:\[9 - 3x - 2x^2\].Rearrange this in descending order of powers of \(x\):\[-2x^2 - 3x + 9\]

Key Concepts

Distributive Property

The Distributive Property is a fundamental principle in algebra that is often used when multiplying two binomials. At its core, this property allows us to distribute or spread a multiplier across terms inside parentheses. When you have an expression like \((a + b)(c + d)\), you apply the Distributive Property by multiplying each term in the first set of parentheses by each term in the second set. This involves pairing each term in the first binomial with every term in the second binomial.For this exercise, consider the expression \((3 - 2x)(3 + x)\):

• Multiply the first terms: \(3 \cdot 3 = 9\).
• Multiply the outer terms: \(3 \cdot x = 3x\).
• Multiply the inner terms: \(-2x \cdot 3 = -6x\).
• Multiply the last terms: \(-2x \cdot x = -2x^2\).

Using the distributive property simplifies these calculations and is essential for working with expressions involving multiple terms.

FOIL Method

The FOIL Method is a specialized application of the Distributive Property used specifically for multiplying two binomials and is an acronym standing for First, Outside, Inside, Last. This method helps ensure that every term from the first binomial is multiplied correctly by every term from the second binomial. It organizes the multiplication process into a sequence that is straightforward to follow.For example, when using the FOIL method on the binomials \((3-2x)(3+x)\):

• First: Multiply the first terms in each binomial, \(3 \cdot 3 = 9\).
• Outside: Multiply the outer terms in the expression, \(3 \cdot x = 3x\).
• Inside: Multiply the inner terms, \(-2x \cdot 3 = -6x\).
• Last: Multiply the last terms in each binomial, \(-2x \cdot x = -2x^2\).

Structured like this, the FOIL method provides a simple and effective way to ensure that all necessary multiplications are performed correctly and is particularly useful for students learning to multiply binomials.

Combining Like Terms

Once you've used the Distributive Property or FOIL Method to multiply binomials, you will often end up with an expression containing several different terms. To simplify this expression, you need to "Combine Like Terms". Like terms are terms whose variables and their exponents match, regardless of the coefficient.For our multiplied binomial example, \(9 + 3x - 6x - 2x^2\),we identify like terms to combine:

• The terms \(3x\) and \(-6x\) both contain \(x\) to the first power. Add these together to simplify: \(3x - 6x = -3x\).

The expression then simplifies to:\[-2x^2 - 3x + 9\].Combining like terms is essential for producing the simplest form of an algebraic expression. It helps to give the result in a clear, clean manner that is easy to interpret, especially when preparing the final answer.