Problem 65 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Apply the Distributive Property

To multiply the binomials \((x + y)(x - 2y)\), we need to apply the distributive property, also known as FOIL (First, Outer, Inner, Last) method. This means we will multiply each term in the first binomial by each term in the second binomial.

Step 2: Multiply the First Terms

Multiply the first terms from each binomial: \(x \times x = x^2\). This gives us the first part of our expression.

Step 3: Multiply the Outer Terms

Now multiply the outer terms: \(x \times -2y = -2xy\).

Step 4: Multiply the Inner Terms

Next, multiply the inner terms: \(y \times x = xy\).

Step 5: Multiply the Last Terms

Finally, multiply the last terms: \(y \times -2y = -2y^2\).

Step 6: Combine All Terms

Add all the terms we found in steps 2 through 5 together to get the expanded form: \(x^2 - 2xy + xy - 2y^2\).

Step 7: Simplify the Expression

Combine like terms in the expression. The terms \(-2xy\) and \(+xy\) can be combined to form \(-xy\). So, the simplified expression becomes \(x^2 - xy - 2y^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic property used to simplify expressions and solve equations. When multiplying a binomial by another binomial, we use this property to distribute each term across the other binomial. In the case of \[(x+y)(x-2y),\]we apply the distributive property to multiply every term in the first binomial, \[(x+y),\]by each term in the second binomial, \[(x-2y).\]This results in multiplying:

• First terms: \(x \times x\)
• Outer terms: \(x \times -2y\)
• Inner terms: \(y \times x\)
• Last terms: \(y \times -2y\)

By performing these calculations, we can effectively distribute all components of the binomials to simplify or expand the expression. This method ensures that no term is left behind in the distribution process, fully expanding the expression into a polynomial ready for further simplification.

FOIL Method

The FOIL method is a popular strategy derived from the distributive property, specifically used for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, representing the order in which you multiply the terms.To multiply \[(x+y)(x-2y),\]you follow these steps:

• **First**: Multiply the first terms in each binomial: \(x \times x = x^2\).
• **Outer**: Multiply the outer terms: \(x \times -2y = -2xy\).
• **Inner**: Multiply the inner terms: \(y \times x = xy\).
• **Last**: Multiply the last terms: \(y \times -2y = -2y^2\).

After obtaining these individual products, add them together to form the expanded expression: \[x^2 - 2xy + xy - 2y^2.\]The FOIL method is a simple, organized way to remember the multiplication sequence, which helps in managing the calculations in a systematic manner and ensuring that all terms are accounted for in the expansion.

Polynomial Simplification

Polynomial simplification involves combining like terms to make an expression easier to read and work with. Once we have expanded the binomials using either the distributive property or FOIL method, we have a polynomial that may contain similar terms.In our example, \[x^2 - 2xy + xy - 2y^2,\]we identify that the terms \(-2xy\) and \(+xy\)are like terms because they both involve the variable combination \(xy\). To simplify this polynomial, combine these like terms:\[-2xy + xy = -xy.\]Thus, our simplified expression becomes: \[x^2 - xy - 2y^2.\]Simplifying polynomials is essential as it reduces the complexity of expressions, making it more straightforward for further operations or evaluations. Clear expression helps in better understanding and efficiency in mathematical computations.