Problem 66 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (y+4)(y+5) $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: (y+4)(y+5). Answer: The simplified algebraic expression is y^2 + 9y + 20.

Step 1: Distribute y from first binomial

Begin by distributing the first term, y, of the first binomial to each term in the second binomial: $$ y(y+5)+ 4(y+5) $$

Step 2: Apply distributive property

Apply the distributive property by multiplying y with each term of the second binomial: $$ (y*y+y*5)+(4*y+4*5) $$

Step 3: Simplify the terms

Continue simplification by simplifying the terms: $$ (y^2+5y)+(4y+20) $$

Step 4: Combine like terms

Combine the like terms (i.e., terms with the same power of y) to get the final simplified expression: $$ y^2+5y +4y+ 20 $$ And the final simplified expression is: $$ y^2+9y+20 $$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It basically means you take a term outside the parentheses and multiply it by each term inside the parentheses. This property is especially useful when dealing with expressions that have variables, like \[(y + 4)(y + 5)\]. Here’s how it works.

• Identify the terms: Let's say you have two binomials like \((a + b)(c + d)\). The distributive property tells us to multiply \(a\) by both \(c\) and \(d\), and then multiply \(b\) by both \(c\) and \(d\).
• Break it down: Apply the distributive property step-by-step. First, distribute \(a\), resulting in \(ac + ad\). Then distribute \(b\), giving \(bc + bd\). Finally, combine all these results: \(ac + ad + bc + bd\).
• Simplify: After using the distributive property, you'll often need to combine like terms to simplify the expression further.

The idea is to re-write the multiplication of binomials in a way that makes it easier to solve by distributing each term. This is just the beginning of unlocking algebraic expressions.

Binomials

A binomial is a type of polynomial with exactly two terms. These terms are usually separated by either a plus (+) or a minus (-) sign. In the expression \((y + 4)(y + 5)\), both \((y + 4)\) and \((y + 5)\) are binomials.

• Structure: A typical binomial looks like \((a + b)\), where \(a\) and \(b\) can be constants, variables, or a combination of both.
• Operations: Binomials can be added, subtracted, or multiplied. Multiplying binomials often requires the use of the distributive property to expand the expression.
• Examples: Some examples of binomials are \((x + 1)\), \((3 - y)\), and \((a + b)\).

Understanding binomials is crucial because they form the building blocks for more complex algebraic expressions. When multiplying two binomials, it's important to ensure all parts are covered in the multiplication process, involving concepts like FOIL (First, Outside, Inside, Last) used in distributing terms.

Simplifying Expressions

Simplifying expressions makes them easier to understand and use. During simplification, we aim to reduce an expression to its simplest form without changing its value. It often involves combining like terms and applying basic arithmetic.

• Like Terms: These are terms that contain the same variables raised to the same power. For example, \(5y\) and \(4y\) are like terms because they both contain \(y\).
• Combining Like Terms: To simplify an expression, combine all like terms. In the expression \(y^2 + 5y + 4y + 20\), \(5y\) and \(4y\) are like terms and can be combined into \(9y\).
• Final Form: After combining like terms, your expression should have as few terms as possible. The expression \(y^2 + 9y + 20\) is the simplest form of the original one.

Through practice, simplification becomes second nature, helping with accuracy and efficiency in solving algebra problems. It's a vital skill for making complex equations manageable.