Problem 46 - College Algebra

Question

For the following exercises, multiply the polynomials. $$ (4 x-6 y)(6 x-4 y) $$

Step-by-Step Solution

Answer

The product is \(24x^2 - 52xy + 24y^2\).

Step 1: Apply the Distributive Property

To multiply the polynomials \((4x - 6y)(6x - 4y)\), we apply the distributive property (FOIL method for binomials). This means we'll multiply each term in the first polynomial by every term in the second polynomial.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \(4x \times 6x\). This gives us: \[ 24x^2 \]

Step 3: Multiply the Outer Terms

Next, multiply the outer terms: \(4x \times -4y\). This calculates to: \[ -16xy \]

Step 4: Multiply the Inner Terms

Now, multiply the inner terms: \(-6y \times 6x\). This results in: \[ -36xy \]

Step 5: Multiply the Last Terms

Finally, multiply the last terms of each binomial: \(-6y \times -4y\). This gives: \[ 24y^2 \]

Step 6: Combine the Terms

Combine all the terms we have calculated: \[ 24x^2 - 16xy - 36xy + 24y^2 \]Combine the like terms \(-16xy\) and \(-36xy\), which results in \(-52xy\). Therefore, the final expression is: \[ 24x^2 - 52xy + 24y^2 \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms in parentheses. In polynomial multiplication, this property is especially useful. It lets us break down complex expressions into simpler parts that are easier to handle.
For instance, in the multiplication problem \((4x - 6y)(6x - 4y)\), the Distributive Property involves distributing each term in the first binomial across all terms in the second binomial. This means each term in the first binomial is multiplied by each term in the second binomial. Doing so ensures that all products are accounted for.
Understanding the Distributive Property lays the groundwork for mastering more advanced algebraic techniques, making it essential for solving polynomial equations efficiently.

FOIL Method

The FOIL Method is a special case of the Distributive Property used specifically for multiplying binomials. Binomials are algebraic expressions that have exactly two terms, such as \((a + b)\) and \((c + d)\). The FOIL method simplifies multiplication by following a straightforward order:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

This ordered approach ensures that no product is overlooked.
In our example, \((4x - 6y)(6x - 4y)\), using FOIL, we would first multiply the first terms \(4x\) and \(6x\) to get \(24x^2\). Then the outer terms \(4x\) and \(-4y\) to get \(-16xy\). Next, the inner terms \(-6y\) and \(6x\) to get \(-36xy\). Lastly, the last terms \(-6y\) and \(-4y\) to get \(24y^2\). The FOIL Method not only helps in organizing these steps but also minimizes the risk of arithmetic errors.

Binomials

Binomials are a type of polynomial with exactly two terms. Each term can be a combination of numbers, variables, or both. Understanding binomials is crucial as they form the basis for much of algebraic operations, including factoring and expanding expressions.
In our multiplication exercise, each polynomial \((4x - 6y)\) and \((6x - 4y)\) is a binomial. When multiplying binomials, it's essential to systematically handle each term to ensure accurate results. Since binomials frequently appear in algebra problems, mastering their manipulation through methods like the FOIL Method is vital.
Being comfortable with binomials allows students to navigate through algebraic challenges more confidently, leveraging their understanding to solve complex equations.

Combining Like Terms

Combining like terms is a process used in algebra to simplify expressions or equations. It involves adding or subtracting terms that have the same variable components raised to the same power.
In our example, after using the FOIL Method, we arrived at the expression \(24x^2 - 16xy - 36xy + 24y^2\). To simplify, we identify like terms. Here, \(-16xy\) and \(-36xy\) are like terms because they share the same variable components \(xy\). By combining these, we get \(-52xy\), thus simplifying the expression to \(24x^2 - 52xy + 24y^2\).
This simplification is crucial for making expressions more manageable and is a key step in solving polynomial equations efficiently. Mastery of this concept helps streamline algebraic processes and prepares students for more advanced topics.