Problem 58 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \(\left(2+y^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(4 + 4y^3 + y^6\).

Step 1: Identify the Special Product Formula

The given expression is \((2 + y^3)^2\). This is a perfect square trinomial, and we can use the square of a binomial special product formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2\) and \(b = y^3\).

Step 2: Apply the Formula

Substitute \(a = 2\) and \(b = y^3\) into the formula: \((2 + y^3)^2 = (2)^2 + 2(2)(y^3) + (y^3)^2\).

Step 3: Calculate Each Term

First, calculate \((2)^2 = 4\). Next, calculate \(2(2)(y^3) = 4y^3\). Finally, calculate \((y^3)^2 = y^6\).

Step 4: Write the Simplified Expression

Combine all terms from the previous step to get the simplified expression. Thus, \((2 + y^3)^2 = 4 + 4y^3 + y^6\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of algebraic expression that results from squaring a binomial. This means you take a binomial, which is an algebraic expression with two terms, and multiply it by itself. The result is a trinomial that is called "perfect" because of its symmetrical pattern. This trinomial takes the form:

• First, the square of the first term.
• Second, twice the product of both terms.
• Third, the square of the second term.

When you identify a perfect square trinomial, it's always helpful to recognize it because it can be expanded or factorized easily using special formulas. The special product formula for a perfect square trinomial is written as \((a + b)^2 = a^2 + 2ab + b^2\). This formula is used extensively in algebra to simplify expressions and solve equations more efficiently.

Square of a Binomial

The square of a binomial refers to multiplying a binomial by itself. The process is quite straightforward when using the special product formula, which simplifies the multiplication. The binomial square formula is expressed as \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a\) and \(b\) are the two terms of your binomial. For example, in the expression \((2 + y^3)^2\), you have two terms: 2 and \(y^3\). By following the formula:

• Calculate \((2)^2\), which gives 4, the square of the first term.
• Calculate \(2 \times 2 \times y^3\), which gives \(4y^3\), twice the product of the two terms.
• Calculate \((y^3)^2\), which gives \(y^6\), the square of the second term.

Combining these parts gives you the expanded form of the binomial squared.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of algebra, representing a wide range of problems and solutions. The main components of algebraic expressions are:

• Numerical Constants: Numbers which tell you how many or how much.
• Variables: Symbols, often letters, that represent unknown values.
• Operations: Addition, subtraction, multiplication, and division.

In the example \((2 + y^3)^2\), this expression consists of a constant (2) and a variable raised to a power (\(y^3\)). When working with algebraic expressions, especially involving special products, it's important to understand the role of each component. This understanding helps you apply formulas correctly and simplify expressions effectively.

Simplification Techniques

Simplification techniques in algebra involve reducing expressions into their simplest form. This makes them easier to work with, especially when solving equations or evaluating expressions. Several key simplification techniques include:

• Combining like terms: Add or subtract terms with the same variables and exponents.
• Using distribution: Use the distributive property to remove brackets and parentheses.
• Applying special product formulas: Simplify expressions by recognizing patterns and using known formulas.

Take the expression \((2 + y^3)^2\). By recognizing it as a square of a binomial, you apply the special product formula to break it down into simpler parts: 4, \(4y^3\), and \(y^6\). Each term is computed separately and then combined into the final simplified form: \(4 + 4y^3 + y^6\). This approach makes it clear and straightforward to manage potentially complex expressions.