Problem 30 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 4(y+3) $$

Step-by-Step Solution

Answer

Question: Perform the multiplication of 4 with the expression (y+3). Answer: 4y + 12

Step 1: Apply the distributive property

To multiply 4 with the expression (y+3), we will use the distributive property. This means we will multiply 4 with each term inside the parentheses: $$ 4(y+3) = 4 * y + 4 * 3 $$

Step 2: Perform the multiplication

Now, let's carry out the multiplication: $$ 4y + 4 * 3 = 4y + 12 $$

Step 3: Combine like terms (if any)

In this case, there are no like terms to combine, so the final expression is: $$ 4y + 12 $$

Key Concepts

Like Terms

In algebra, the concept of "like terms" is crucial as it allows for the simplification of expressions. Like terms are terms in an expression that have the exact same variable part. This means they have the same variable raised to the same power, though their coefficients can be different.

For example, in the expression \(3x + 5x\), both terms have the variable \(x\) with the same exponent (which is 1, here implied). Hence, they are like terms and can be combined to give \(8x\).

• Term: A single mathematical expression, which could be a number, a variable, or a combination of both through multiplication or division.
• Like Terms: Terms that have identical variable components (including their exponents), allowing for combination through addition or subtraction.
• Unlike Terms: Terms that do not have identical variable parts or powers.

In our exercise, after performing multiplication, we got \(4y + 12\). Here, \(4y\) and \(12\) have different variable parts, meaning there are no like terms to combine.'

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial across every term of another. The distributive property is often utilized in this process to simplify the multiplication and ensure every term is accounted for. In simpler terms, each term in the first polynomial multiplies every term in the second.

In the provided exercise, we used the distributive property as a core method of multiplying the polynomial \(4(y + 3)\). You multiply the 4 by each term inside the parentheses:

• Multiply 4 by \(y\), resulting in \(4y\).
• Multiply 4 by 3, resulting in 12.

After distributing, you add the results: \(4y + 12\). This method allows each term to be correctly multiplied and properly accounted for, ensuring the expression is simplified correctly.

Algebraic Expressions

Algebraic expressions form the foundational building blocks in algebra, representing quantities through variables and constants combined using mathematical operations. They can range from simple forms with just one term, like \(5x\), to complex multi-term polynomials such as \(3x^2 + 2x + 1\).

Here are some essential components of algebraic expressions:

• Variables: Symbols (like \(x\), \(y\), or \(z\)) that represent unknown or varying numbers.
• Constants: Fixed values, such as 1, 2, or 3, which do not change.
• Coefficients: Numbers multiplying the variables, for instance, 4 in \(4y\).

Algebraic expressions allow the representation of real-world situations mathematically, enabling analysis, simplification, and problem-solving. In the exercise \(4(y + 3)\), we started with a simple expression which was expanded using the distributive property into the polynomial \(4y + 12\). Understanding such expressions is pivotal for algebraic manipulation and further mathematical exploration.