Problem 50 - Algebra 2

Question

Simplify. $$ (y+5)(y-3) $$

Step-by-Step Solution

Answer

The simplified expression is \( y^2 + 2y - 15 \).

Step 1: Apply the Distributive Property

The distributive property, also known as the FOIL method for binomials (First, Outer, Inner, Last), helps us to multiply two binomials. Start by multiplying the first terms together, then the outer terms, followed by the inner terms, and finally the last terms. Keep the operations in mind: \[ (y+5)(y-3) = y \times y + y \times (-3) + 5 \times y + 5 \times (-3) \] This results in:\[ y^2 - 3y + 5y - 15 \]

Step 2: Combine Like Terms

The expression \( y^2 - 3y + 5y - 15 \) has like terms that can be combined. Specifically, the terms \(-3y\) and \(+5y\) are both coefficients of \(y\), and should be added together. \[ -3y + 5y = 2y \] Replacing \(-3y + 5y\) with \(2y\) gives us:\[ y^2 + 2y - 15 \]

Step 3: Write the Simplified Expression

After combining like terms, the expression is simplified to:\[ y^2 + 2y - 15 \]Now, the expression is fully simplified and no further combination of terms is possible.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle crucial for simplifying expressions. By distributing or spreading multiplication over addition or subtraction, this property allows complex equations to be broken down into simpler parts. For example, when you have an expression like

• \((y+5)(y-3)\)

you can apply the distributive property to handle the multiplication of each term from the first binomial with every term from the second binomial.
In our example, this means:

• Mulitply \(y\) by \(y\) to get \(y^2\)
• Mulitply \(y\) by \(-3\) to get \(-3y\)
• Mulitply \(5\) by \(y\) to get \(5y\)
• Mulitply \(5\) by \(-3\) to get \(-15\)

This step is essentially a way to ensure that each term in the first binomial interacts properly with each term in the second. By multiplying each of these pairs and summing the results, you arrive at an expression before simplification: \[ y^2 - 3y + 5y - 15 \]In essence, the distributive property provides a methodical way to tackle expressions where multiplication is involved within parentheses, setting a strong foundation for further simplification.

FOIL Method

The FOIL method is a specialized application of the distributive property for handling the multiplication of two binomials. It's an acronym that stands for First, Outer, Inner, Last, describing the order in which you multiply terms from two binomials. Let's break it down using

• \((y + 5)(y - 3)\)

• **First:** Multiply the first terms of each binomial: \(y \times y = y^2\)
• **Outer:** Multiply the outer terms: \(y \times (-3) = -3y\)
• **Inner:** Multiply the inner terms: \(5 \times y = 5y\)
• **Last:** Multiply the last terms: \(5 \times (-3) = -15\)

FOIL helps streamline the multiplication of binomials, ensuring that all possible pairs are covered. This method is particularly useful for students who might find direct distribution challenging as it provides a structured approach. The resulting expression after applying FOIL is: \[ y^2 - 3y + 5y - 15 \]The clarity provided by FOIL makes it invaluable in understanding how elements combine during multiplication. It's a key technique in polynomial simplification that helps minimize errors through clear, ordered steps.

Combining Like Terms

Combining like terms is a vital process in algebra that simplifies expressions further by consolidating terms with identical variables raised to the same power. Once you've expanded an expression using the FOIL method, the next step is to tidy it up by combining any similar items. For instance, in the expression

• \( y^2 - 3y + 5y - 15 \)

you need to identify which terms share the same variables and powers. In this example:

• The terms \( -3y \) and \( +5y \) can be combined, because both are single \(y\) terms.

When we add these together, we calculate \( -3y + 5y = 2y \). Rewriting the expression, you now have:

• \( y^2 + 2y - 15 \)

This technique of combining like terms streamlines your multivariate polynomial into its simplest form, making it both easier to understand and work with. Once the like terms are aggregated, the polynomial is said to be simplified, completing the process of polynomial simplification.