Problem 50 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ y(y+7) $$

Step-by-Step Solution

Answer

Question: Multiply y with the expression (y+7) and simplify the result. Answer: The product of y and (y+7) is \(y^2 + 7y\).

Step 1: Identify the expressions

In the given problem, we need to multiply y with the expression (y+7).

Step 2: Perform the multiplication

To multiply y with (y+7), use the distributive property, which states that \(a (b + c) = ab + ac\). Here, \(a = y\), \(b = y\), and \(c = 7\). Now multiply y with both the terms inside the parentheses: $$ y(y+7) = y \cdot y + y \cdot 7 $$

Step 3: Calculate the products

Now, calculate the products: $$ y \cdot y + y \cdot 7 = y^2 + 7y $$

Step 4: Combine like terms if present

In the expression \(y^2 + 7y\), there are no like terms to be combined. So the final result of the multiplication is: $$ y(y+7) = y^2 + 7y $$

Key Concepts

Multiplication of Polynomials

Multiplication of polynomials involves using a technique called the Distributive Property, which helps to simplify algebraic expressions by breaking them down into smaller, manageable parts. When multiplying a polynomial by a monomial, such as in the expression \(y(y+7)\), the distributive property tells us to multiply the single term outside the parentheses, in this case \(y\), by each term within the parentheses.

Here's a quick look at the steps:

• Identify Your Terms: First, recognize the components involved. When you have \(y(y+7)\), you identify \(y\) as the term you will distribute. Inside the parentheses, you have \(y\) and \(7\).
• Distribute: Apply the distributive property: \(a(b + c) = ab + ac\). Here, distribute \(y\) to both \(y\) and \(7\). This results in \(y \cdot y + y \cdot 7\).
• Calculate the Products: Perform the multiplication for each term: \(y \cdot y = y^2\) and \(y \cdot 7 = 7y\).

As you can see, by systematically applying each step, the multiplication of polynomials becomes easier and clearer, resulting in the expression \(y^2 + 7y\).

Combining Like Terms

Combining like terms is a crucial step in simplifying polynomials. Like terms are terms that have the same variable raised to the same power. In simpler terms, these are terms that contain the exact same variables and can thus be combined to simplify calculations. For example, in expressions such as \(3x + 5x\), both terms can be combined because they are 'like' — their variable part, \(x\), is the same.

When addressing the expression \(y^2 + 7y\):

• Identify Like Terms: In the expression, \(y^2\) and \(7y\) do not share the same variable raised to the same power.


• No Combination Needed: Since \(y^2\) and \(7y\) are not like terms, they are left as they are. If there were multiple terms with \(y\) or \(y^2\), you would add or subtract their coefficients.

This process is about ensuring that the expression is in its simplest, most concise form, which is particularly useful for later operations or solving equations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols such as plus or minus signs. It is important to understand that these expressions represent quantities that can vary or change depending on the values of the variables involved. For example, \(y + 7\) is an algebraic expression where \(y\) is a variable, and \(7\) is a constant.

Key components of understanding algebraic expressions include:

• Variables: Symbols representing unknown or variable quantities. In \(y + 7\), \(y\) is the variable.
• Constants: Numbers with fixed values. The number \(7\) in the expression is a constant.
• Operations: Symbols that dictate the mathematical operations to be performed. In this case, addition is the operation.

Algebraic expressions can become quite complex with more variables and operations, but breaking them down into these components helps in performing operations such as multiplication and addition, making them easier to manage and solve.