Problem 104 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (2 y+5)(4 y+5) $$

Step-by-Step Solution

Answer

Answer: \(8y^2 + 30y + 25\)

Step 1: Distribute the first term

First, distribute the first term in the first expression, which is \(2y\). So, multiply \(2y\) with both terms in the second expression \((4y + 5)\). We get: $$ (2y)(4y) + (2y)(5) $$

Step 2: Distribute the second term

Next, distribute the second term from the first expression, which is \(5\). Multiply \(5\) with both terms in the second expression \((4y + 5)\). We get: $$ (5)(4y) + (5)(5) $$

Step 3: Combine the results

Now, combine the results from Steps 1 and 2 to obtain the expression: $$ (2y)(4y) + (2y)(5) + (5)(4y) + (5)(5) $$

Step 4: Perform multiplication

Next, perform the multiplication for each term in the expression: $$ 8y^2 + 10y + 20y + 25 $$

Step 5: Combine like terms

Finally, combine the like terms in the expression. In this case, we have two like terms (\(10y\) and \(20y\)). Add them together to get: $$ 8y^2 + 30y + 25 $$ The simplified algebraic expression is \(8y^2 + 30y + 25\).

Key Concepts

Distribution Method

The distribution method is a crucial technique in algebra that involves multiplying each term in one expression by every term in another. This process is often used when dealing with expressions that are multiplied together, such as

• polynomials or
• factored expressions.

In our exercise, the distribution method was applied to the expression \((2y + 5)(4y + 5)\).

We distributed each term from the first polynomial by multiplying it with every term in the second polynomial. This step-by-step approach ensures no term is left unmultiplied.

The process can be broken down into simpler parts:

• Take the first term from the first polynomial, which is \(2y\).
• Multiply \(2y\) with each term in the second polynomial \((4y + 5)\).
• Then repeat the same distribution process with the second term of the first polynomial (which is a constant 5).

This method reveals all interactions between terms in the two expressions, setting the stage for further simplification.

Combining Like Terms

Once distribution is complete, the next step is to combine like terms. This means grouping and simplifying terms in the expression that have the same variable raised to the same power. Identifying like terms is essential for reducing expressions to their simplest form.

In our exercise, after applying the distribution method, we ended up with:

• \(8y^2\)
• \(10y\)
• \(20y\)
• \(25\)

Here, \(10y\) and \(20y\) are like terms because they both contain the same variable, \(y\), raised to the first power.

To combine them, simply add their coefficients:\(10y + 20y = 30y\).Finally, the expression becomes \(8y^2 + 30y + 25\).This combining process is essential for reaching the final simplified expression.

Polynomial Simplification

Simplifying polynomials targets creating the simplest form of expression possible. This process involves applying both the distribution method and the combining of like terms.

After completing these initial steps, ensure there are no more like terms to combine or similar operations left undone. This forms a comprehensive view of the mathematical relationship.

Our final expression, \(8y^2 + 30y + 25\), exemplifies a fully simplified polynomial. It includes:

• No like terms remain unsimplified, making operations straightforward and minimal.
• The expression is concise, providing clearer insight into the relationships among the terms.

Achieving polynomial simplification aids in tackling more complex algebraic problems. It creates an easy pathway to solving equations and can simplify further substitutions or mathematical manipulations. Recognizing and remembering these key steps supports proficiency in simplifying a wide array of algebraic expressions.