Problem 53 - Algebra A Combined Function

Question

Multiply. $$ (y+4)(y+4) $$

Step-by-Step Solution

Answer

The product is \(y^2 + 8y + 16\).

Step 1: Identify the Expression

We are given the expression \((y+4)(y+4)\). This is a binomial squared, which means we are multiplying the same binomial by itself.

Step 2: Apply the Formula for Squaring a Binomial

The square of a binomial \((a+b)^2\) is given by the formula \(a^2 + 2ab + b^2\). Here, \(a = y\) and \(b = 4\).

Step 3: Compute \(a^2\)

Calculate \(y^2\). This is simply \(y \times y = y^2\).

Step 4: Compute \(2ab\)

Compute \(2 \times y \times 4\). This gives us \(8y\).

Step 5: Compute \(b^2\)

Calculate \(4^2\). This results in 16.

Step 6: Combine the Terms

Add the terms together: \(y^2 + 8y + 16\).

Key Concepts

Multiplication of Polynomials

When you multiply polynomials, you're essentially multiplying terms of one polynomial by the terms of another and then combining like terms. This is a foundational skill in algebra, allowing you to simplify expressions that might seem overwhelming at first. Common methods include:

• Distributive Property: Each term in the first polynomial is multiplied by each term in the second polynomial. This is particularly useful for lower degree polynomials.
• FOIL Method: Specifically used for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, representing the terms you need to multiply together.

After performing these multiplications, it's important to combine like terms, which are terms with the same variables raised to the same powers. This step ensures the polynomial is in its simplest form.

Squaring Binomials

The process of squaring a binomial involves finding the square of a two-term expression. It is different from simply multiplying each term by 2. Instead, you need to apply the formula: \((a + b)^2 = a^2 + 2ab + b^2\).This means:

• You square the first term, \(a\), giving \(a^2\).
• Multiply the two terms together and then by 2, giving \(2ab\).
• Square the second term, \(b\), resulting in \(b^2\).

For example, squaring \((y + 4)\) begins with identifying \(a = y\) and \(b = 4\). Applying the formula, you get:

• \(a^2 = y^2\)
• \(2ab = 8y\)
• \(b^2 = 16\)

Combining these gives you the squared expression: \(y^2 + 8y + 16\). This strategy ensures correctness and saves time in polynomial expansion.

Algebraic Expressions

Algebraic expressions combine numbers and variables with operations such as addition, subtraction, multiplication, and division. They are fundamental in understanding algebra because:

• They represent real-world quantities.
• They allow for the generalization of arithmetic taught in basic mathematics.

An algebraic expression's complexity can vary based on the number of terms and the operations present. For example:

• A simple expression like \(x + 3\) is a basic algebraic expression.
• More complex expressions include multiple terms and operations, like \(3x^2 - 5x + 2\).

Understanding how to manipulate these expressions, such as through multiplication of polynomials or squaring binomials, helps solve equations and identify relationships between variables. It's a way of setting up and solving real-life problems using mathematics. Thus, mastering the use of these expressions is key in progressing your math skills.