Problem 67 - Algebra A Combined Function

Question

Multiply. \((3 y+5)^{2}\)

Step-by-Step Solution

Answer

The result is \(9y^2 + 30y + 25\).

Step 1: Understand the Problem

We need to multiply the expression \((3y+5)^2\). This is a binomial raised to the power of 2.

Step 2: Use the Binomial Square Formula

The binomial square formula is \((a+b)^2 = a^2 + 2ab + b^2\). In our case, we identify \(a = 3y\) and \(b = 5\).

Step 3: Compute \(a^2\)

Calculate \((3y)^2\). This is computed as \((3^2)\times (y^2) = 9y^2\).

Step 4: Compute \(2ab\)

Calculate \(2 \times 3y \times 5 = 30y\). Here, we multiply the terms according to the highlighted formula.

Step 5: Compute \(b^2\)

Calculate \(5^2 = 25\). This squares the second term in the binomial.

Step 6: Combine the Terms

Finally, combine all the components together: \(9y^2 + 30y + 25\). This gives us the expanded form.

Key Concepts

Binomial Square Formula

The Binomial Square Formula is a special algebraic identity that simplifies the process of squaring a binomial. A binomial is simply a two-term algebraic expression such as \((a + b)\). When squaring a binomial, you end up with three terms: two square terms and one double product term. This formula helps to streamline the multiplication process by expressing it as:

• \((a+b)^2 = a^2 + 2ab + b^2\)

This means that you take the square of the first term, add twice the product of the two terms, followed by the square of the second term.

Using this formula can save time, especially in complex calculations or when dealing with polynomials that have more terms. It ensures accuracy and helps to understand the intricate relationships between the terms after expansion.

For example, in the case of the expression \((3y+5)^2\), the formula deconstructs the problem into smaller parts, allowing us to focus individually on squaring \(3y\), multiplying \(2\cdot3y\cdot5\), and squaring \(5\). By following these steps, we achieve an accurate expanded polynomial.

Polynomial Expansion

Polynomial Expansion is the process of expressing a compressed polynomial expression as a sum of terms. When we expand a polynomial, we take an expression involving factors that haven't been multiplied together and express it as their product.

In the context of binomial expressions, such as \((a+b)^2\), expansion means distributing each term within the parentheses and simplifying them further. The goal is to bring out each term separately so that the expression is written in a standard polynomial form.

• The expanded form usually clarifies the degree of the polynomial, which is the highest power of the variable within the polynomial.
• In practical applications, polynomial expansion is useful for identifying coefficients and powers that play crucial roles in graphing polynomial functions, among other tasks.

For the binomial \((3y+5)^2\), expansion leads to a standard polynomial form: \(9y^2 + 30y + 25\). This expression clearly shows each term's power and coefficient, which provides deeper insight into its characteristics.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can consist of numbers, variables, and operations. They are fundamental components in algebra, and understanding how to manipulate them is crucial for solving equations and simplifying expressions.

An algebraic expression can take different forms, including polynomials, which are sums of multiple terms such as \(ax^n + bx^{n-1} + \, \ldots\, + c\). Binomial expressions are a specific kind of polynomial with exactly two terms, like \((3y + 5)\) in our exercise.

• Variables, like \(y\), serve as placeholders that can represent unknown values or quantities that can change.
• Understanding how to work with expressions, especially binomials and polynomials, is critical for conducting algebraic operations such as factoring, expanding, or solving equations.

By recognizing and understanding the parts of an algebraic expression, such as the coefficients and variables, students can handle more complex algebraic challenges. It's important to practice manipulating algebraic expressions to gain confidence in areas like polynomial expansion and simplification.