Problem 38 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((y-1)(y+5)\)

Step-by-Step Solution

Answer

The simplified expression is \(y^2 + 4y - 5\).

Step 1: Apply FOIL Method

FOIL stands for First, Outer, Inner, Last. We'll use this method to expand the expression. First, multiply the first terms:\(y imes y = y^2\).

Step 2: Multiply Outer Terms

Now, multiply the outer terms:\(y imes 5 = 5y\).

Step 3: Multiply Inner Terms

Next, multiply the inner terms:\(-1 imes y = -y\).

Step 4: Multiply Last Terms

Finally, multiply the last terms:\(-1 imes 5 = -5\).

Step 5: Combine the Terms

Now combine all the terms from the previous steps:\(y^2 + 5y - y - 5\).

Step 6: Simplify the Expression

Combine like terms to simplify the expression:\(y^2 + (5y - y) - 5 = y^2 + 4y - 5\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations. They are foundational in algebra and crucial for understanding and mastering various mathematical concepts. In the expression \((y-1)(y+5)\), each part of the expression is linked by multiplication, with 'y' being the variable go-between in both terms. Here’s why each component matters:

• Variables act as placeholders for numbers and can represent different values. In our example, 'y' is the variable.
• Constants are fixed values. In the terms \(-1\) and \(5\), these numbers are constants.
• Operations are used to combine these constants and variables. Typically, you're adding, subtracting, multiplying, and dividing.

Algebraic expressions are powerful tools. They allow us to communicate complex mathematical ideas succinctly.

Multiplication of Expressions

Multiplying algebraic expressions is a method used to find the product of two or more expressions. This is where the FOIL method shines. FOIL, standing for First, Outer, Inner, Last, helps simplify multiplication of two binomials, as seen in \((y-1)(y+5)\). Here's how it works:

• **First Terms:** Multiply the first terms from each binomial: \(y \times y = y^2\). This gives you the leading term of the new expression.
• **Outer Terms:** Next, multiply the outer terms, treating the expressions as ends: \(y \times 5 = 5y\). This begins forming the middle terms.
• **Inner Terms:** Then, multiply the inner terms which are closer to each other: \(-1 \times y = -y\). This adds further to those middle terms.
• **Last Terms:** Lastly, multiply last terms: \(-1 \times 5 = -5\). This concludes the formation of your expression.

These steps ensure every part of each binomial is considered, resulting in a complete product expression you can further simplify.

Simplifying Expressions

Simplifying expressions involves combining like terms in a polynomial to present it in its simplest form. Once you've expanded an expression, like we did with the FOIL method, the next step is to simplify it. The aim is to make it more manageable and easier to understand. Take your expanded expression \(y^2 + 5y - y - 5\):

• **Identify Like Terms:** In this case, \(5y\) and \(-y\) are like terms because they contain the same variable 'y'.
• **Combine Like Terms:** Add or subtract like terms. For our expression, combine \(5y - y = 4y\).
• **Rewrite the Expression:** After combining, your expression simplifies to \(y^2 + 4y - 5\).

Simplifying makes complex expressions easier to work with and can help solve equations more efficiently. It's a key skill in algebra that aids in clearer problem-solving.