Problem 54 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 4 y(y+7) $$

Step-by-Step Solution

Answer

Answer: The product is $$4y^{2} + 28y$$.

Step 1: Identify the monomial and binomial

In the given expression, the monomial is $$4y$$ and the binomial is $$(y + 7)$$.

Step 2: Use the distributive property

Distribute the monomial $$4y$$ to each term inside the binomial parenthesis: $$4y(y+7) = 4y \cdot y + 4y \cdot 7$$

Step 3: Carry out the multiplication

Now multiply the monomial with the individual terms: $$4y \cdot y = 4y^{2}$$ and $$4y \cdot 7 = 28y$$

Step 4: Combine the resulting terms

Put the multiplied terms together: $$4y^{2} + 28y$$ Since there are no like terms to combine, the result is the expanded expression: $$4y^{2} + 28y$$

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used when dealing with expressions involving both monomials and binomials. It states that a term outside a parenthesis can be distributed or multiplied by every term inside the parenthesis. This principle helps to simplify expressions and solve equations efficiently.
To illustrate, consider the expression \(4y(y+7)\). Here, the distributive property allows us to multiply the monomial \(4y\) by each term inside the binomial \((y + 7)\).

• First, multiply \(4y\) by \(y\) to get \(4y^2\).
• Then, multiply \(4y\) by \(7\) to obtain \(28y\).

By applying the distributive property, you can transform the expression into \(4y^2 + 28y\).
This method is not only useful for multiplication but also when factoring, as it allows you to break down and simplify algebraic expressions.

Combining Like Terms

Combining like terms is an essential process in simplifying algebraic expressions. Like terms are terms that have identical variable parts raised to the same power. This means you can add or subtract their coefficients while keeping the variable part unchanged.
In our problem, the expanded expression is \(4y^2 + 28y\). Before combining like terms, identify any terms with the same variables and exponents:

• \(4y^2\) is a term by itself, as it represents \(y\) raised to the second power.
• \(28y\) is another term representing \(y\) raised to the first power.

Here, there are no like terms to combine because the exponents of \(y\) in each term are different.
Therefore, the expression remains as \(4y^2 + 28y\). This step ensures that your expression is as simplified as possible, making it easier to understand and use in further calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra, representing quantities in a compact, symbolic form.
An expression can be as simple as a single variable or as complex as multiple terms combined by addition, subtraction, multiplication, and division.
In the expression \(4y(y+7)\), there are key components:

• **Monomial**: A single term, like \(4y\), made up of a coefficient and a variable.
• **Binomial**: Two terms added or subtracted, such as \(y + 7\).

The goal is often to simplify these expressions or manipulate them to solve problems. This involves understanding how to perform operations like distribution and combining like terms, as explained in previous sections.
Simplifying algebraic expressions makes equations more manageable and helps uncover relationships between variables, which is at the heart of solving many algebra problems.