Problem 59 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The product is \(-2x^2 + 7x - 6\).

Step 1: Apply the Distributive Property

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

Step 2: Multiply the First Terms

Multiply the first term of each binomial: \(-2x \times x = -2x^2\).

Step 3: Multiply the Outer Terms

Next, multiply the outer terms of the binomials: \(-2x \times -2 = 4x\).

Step 4: Multiply the Inner Terms

Multiply the inner terms: \(3 \times x = 3x\).

Step 5: Multiply the Last Terms

Finally, multiply the last terms of each binomial: \(3 \times -2 = -6\).

Step 6: Combine Like Terms

Add all the results from the previous steps: \(-2x^2 + 4x + 3x - 6\). Combine the like terms \(4x\) and \(3x\) to get: \(-2x^2 + 7x - 6\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps break down complex expressions into simpler parts. It provides a way to multiply a single term by a group of terms inside parentheses. This property is defined as:

• For any numbers or variables, a, b, and c:
  • \(a(b + c) = ab + ac\)

When dealing with binomials, the distributive property ensures that each term in one binomial is multiplied by every term in the other. This approach is not just limited to algebraic expressions, but is pretty handy in various mathematical calculations. By systematically distributing the terms, it simplifies the multiplication process, making it easier and more manageable to solve.

The FOIL Method Demystified

The FOIL method is a particular application of the distributive property that deals specifically with multiplying two binomials. The term FOIL is an acronym that stands for:

• F - First (Multiply the first terms of each binomial)
• O - Outer (Multiply the outer terms in the expression)
• I - Inner (Multiply the inner terms)
• L - Last (Multiply the last terms of each binomial)

The FOIL method is a straightforward and systematic way to ensure that every pair of terms is accounted for. For the expression \((-2x + 3)(x - 2)\), applying foil involves:

• First: \(-2x imes x = -2x^2\)
• Outer: \(-2x imes -2 = 4x\)
• Inner: \(3 imes x = 3x\)
• Last: \(3 imes -2 = -6\)

By following these steps, you reach the expression \(-2x^2 + 4x + 3x - 6\). This method ensures no term is missed during multiplication.

Combining Like Terms

After multiplying the terms using the distributive property or FOIL method, the next crucial step is combining like terms. Like terms are terms that have the same variable raised to the same power. Only the coefficients (numbers in front of variables) add up or subtract.

• For example, in \(-2x^2 + 4x + 3x - 6\), both \(4x\) and \(3x\) are like terms because they contain the same variable\(x\).

To combine them, add their coefficients: \(4 + 3\). This result gives \(7x\), simplifying the expression to \(-2x^2 + 7x - 6\).
It is essential to identify and combine like terms as it simplifies the expression, giving a cleaner solution.

Exploring Algebraic Expressions

Algebraic expressions encompass variables, numbers, and operations such as addition and multiplication. These are statements of equality or inequality, comprised of one or more terms. Each term is a product of constant (number) and variable(s).
Algebraic expressions can be as simple as a sole variable or number \(x, 5\), or more complex like \(-2x^2 + 7x - 6\).

• Within expressions, identifying the structure helps in simplifying them and solving any equation they might be part of.
• By understanding addition and multiplication within expressions, you can solve many algebra problems.
• Remember, both the distributive property and FOIL method help deal with algebraic expressions by simplifying multiplication.

Overall, grasping the basics of algebraic expressions is crucial for mastering more advanced algebraic concepts.