Problem 11 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (y+3)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression (y+3)^2 is y^2 + 6y + 9.

Step 1: (1) Expand using the distributive property (FOIL method)

We'll start by applying the FOIL method to expand the given expression. \((y+3)^{2}\) can also be written as \((y+3)(y+3)\). Applying FOIL, we get: First: \(y * y = y^2\) Outer: \(y * 3 = 3y\) Inner: \(3 * y = 3y\) Last: \(3 * 3 = 9\) Now, we will combine these terms together.

Step 2: (2) Combine like terms

In this step, we will add the terms from the previous step together, and combine any like terms: \(y^2 + 3y + 3y + 9\) Notice that the two middle terms, \(3y\) and \(3y\), are like terms since they both have the variable \(y\). We can combine these terms by adding their coefficients: \(y^2 + (3 + 3)y + 9\)

Step 3: (3) Write the simplified expression

After combining the like terms, our simplified expression is: \(y^2 + 6y + 9\) So, the simplified algebraic expression for \((y+3)^{2}\) is \(\boxed{y^2 + 6y + 9}\).

Key Concepts

Distributive Property

The distributive property is fundamental in simplifying algebraic expressions. It involves spreading out multiplication over addition or subtraction within parentheses. For instance, in the expression \( a(b + c) \), the distributive property allows us to multiply 'a' by both 'b' and 'c' individually. The result is \( ab + ac \). This property is especially useful when dealing with binomials or larger expressions.

Using the distributive property helps to expand expressions so that they can be simplified further, allowing us to combine like terms and solve equations more efficiently. Remember, the distributive property is like sharing: whatever is outside the parentheses must be multiplied by everything inside.

FOIL Method

The FOIL method is a specific application of the distributive property used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, which represents the order in which we multiply the terms. If you have \( (x + y)(a + b) \), following the FOIL sequence, you'll get:

• First: Multiply the first terms in each binomial (\( x * a \)).
• Outer: Multiply the outer terms (\( x * b \)).
• Inner: Multiply the inner terms (\( y * a \)).
• Last: Multiply the last terms (\( y * b \)).

Once all the terms have been multiplied, the next step is to combine any like terms to simplify the expression further.

It is an efficient way to handle expressions like \( (y + 3)^2 \) without missing any terms. Students can benefit from mastering the FOIL method, as it makes tackling complex polynomial multiplications much easier.

Combine Like Terms

Once you've expanded an algebraic expression, combining like terms is the next step towards simplification. Like terms are terms that have the same variables raised to the same powers, although their coefficients can be different. For example, in \( 5x + 3 - 2x + 8 \) the terms \( 5x \) and \( -2x \) are like terms, and so are \( 3 \) and \( 8 \).

You can combine like terms by adding or subtracting their coefficients while keeping the variable part the same. In the expression above, combining like terms would give us \( (5 - 2)x + (3 + 8) \) which simplifies to \( 3x + 11 \). This process reduces the number of terms in the expression and often leads to a much cleaner, simpler form.

Algebraic Expansion

Algebraic expansion is the process of removing parentheses from an algebraic expression and simplifying it as much as possible. This includes using the distributive property and the FOIL method for binomials, as well as combining like terms. Expansion allows us to transform complicated expressions into a more straightforward form that is easier to use in subsequent calculations.

Consider the expression \( (2x + 3)(x - 5) \). To expand it, we would apply the FOIL method: \( 2x \times x - 2x \times 5 + 3 \times x - 3 \times 5 \), which gives us \( 2x^2 - 10x + 3x - 15 \). Then we combine like terms to reach the simplified form \( 2x^2 - 7x - 15 \). Always ensure each term is addressed during expansion to avoid errors in the final answer.