Problem 55 - College Algebra with Modeling and Visualization

Question

Multiply the binomials. $$(y+5)(y-7)$$

Step-by-Step Solution

Answer

The product is \(y^2 - 2y - 35\).

Step 1: Apply the Distributive Property

To multiply the binomials \((y+5)(y-7)\), use the distributive property, often referred to as the FOIL method (First, Outside, Inside, Last). Begin by multiplying the first terms of each binomial: \(y \cdot y = y^2\).

Step 2: Multiply the Outside Terms

Next, multiply the outside terms of the binomials: \(y \cdot (-7) = -7y\).

Step 3: Multiply the Inside Terms

Now multiply the inside terms: \(5 \cdot y = 5y\).

Step 4: Multiply the Last Terms

Finally, multiply the last terms of each binomial: \(5 \cdot (-7) = -35\).

Step 5: Combine All Products

Add up all the products from the previous steps: \(y^2 - 7y + 5y - 35\).

Step 6: Simplify the Expression

Combine the like terms, which are the linear terms \(-7y + 5y\). This simplifies to \(-2y\). Therefore, the expression becomes \(y^2 - 2y - 35\).

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to multiply a single term by terms inside a parenthesis. For binomial multiplication like \((y+5)(y-7)\), this property distributes each term of one binomial across each term of the other.
In our example, the distribution involves each term of the first binomial \((y+5)\) being multiplied by each term of the second binomial \((y-7)\).

• First, multiply the terms within the binomials, starting with the terms outside and then those inside.
• This results in products that represent sections of the final expanded polynomial.

The ability to distribute effectively is crucial for simplifying expressions and solving equations.

Foiling Method

The FOIL method is a specific strategy to correctly apply the distributive property for binomials. FOIL stands for:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the binomials.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

By following these orderly steps, we ensure no terms are missed in the multiplication process. This method is a quick and reliable way to remember the steps for binomial multiplication, leading us directly to the complete expansion.

Polynomial Arithmetic

Polynomial arithmetic involves operations like addition, subtraction, multiplication, and division on polynomials. When dealing with the multiplication of binomials like \((y+5)(y-7)\), we are essentially dealing with a form of polynomial arithmetic.
Once we distribute the terms using the FOIL method, we must simplify the resulting expression by combining like terms in our polynomial.

• Combine Like Terms: Here, the terms \(-7y\) and \(+5y\) combine to form \(-2y\).
• These like terms help reduce the expression to its simplest form.

Understanding polynomial arithmetic is essential for simplifying expressions and solving polynomial equations.