Problem 57 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \(\left(x^{2}+1\right)^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(x^4 + 2x^2 + 1\).

Step 1: Identify the Expression

The expression given is \((x^2+1)^2\). We need to recognize that it is a binomial squared, which can be expanded using a Special Product Formula.

Step 2: Apply the Binomial Square Formula

Recall that the formula for the square of a binomial is \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^2\) and \(b = 1\).

Step 3: Expand the Expression

Using the binomial square formula, expand \((x^2+1)^2 = (x^2)^2 + 2(x^2)(1) + 1^2\).

Step 4: Simplify Each Term

Calculate each term: 1. \((x^2)^2 = x^4\)2. \(2(x^2)(1) = 2x^2\)3. \(1^2 = 1\)

Step 5: Combine Like Terms

Combine all the simplified terms: \(x^4 + 2x^2 + 1\). This is the expanded and simplified form of the expression.

Key Concepts

Binomial Expansion

Binomial expansion is a handy technique used in algebra to expand expressions that are raised to a power. Understanding this concept is crucial when dealing with polynomial equations. It involves using a known formula to expand the expression quickly. Specifically, when dealing with expressions in the form of \((a + b)^n\), the expansion can easily be carried out using the formulas and techniques associated with binomials.

For instance, the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\) allows us to expand a simple binomial raised to the power of two quickly. The trick lies in identifying the components \(a\) and \(b\) from the expression and applying them to the formula like a recipe.

In our example, the expression \((x^2+1)^2\) is recognized as the square of the binomial. Using the formula, it enables us to express it as \((x^2)^2 + 2(x^2)(1) + 1^2\). This expansion simplifies our problem dramatically and efficiently.

Algebraic Expressions

Algebraic expressions form the foundation for higher-level mathematics operations. They consist of variables, coefficients, and constants linked through operations like addition, subtraction, multiplication, and even division. Mastery of algebraic expressions allows students to manipulate and solve equations of varying complexity.

In our problem, the algebraic expression \((x^2 + 1)^2\) is a higher-level expression that includes a polynomial component \(x^2 + 1\). Breaking it down involves identifying different parts of the expression, such as recognizing terms \(x^2\) and \(1\).

• **Variables**: Symbols, like \(x\), representing numbers that can vary.
• **Coefficients**: Fixed numbers (integer, fraction, etc.) that multiply a variable, here we can consider \(2\) as a coefficient during expansion.
• **Constants**: The fixed numbers present in an expression, such as \(+1\).

Algebraic expressions must often be simplified, as shown in our example, where we took the expanded form and combined like terms to arrive at the simplest form, \(x^4 + 2x^2 + 1\).

Polynomial Simplification

Simplifying polynomials involves reducing algebraic expressions into their simplest form while still maintaining their equality to the original expression. This process ensures that the polynomial is presented in the most efficient manner, revealing essential characteristics for further mathematical operations.

For example, after expanding \((x^2 + 1)^2\) into \((x^2)^2 + 2(x^2)(1) + 1^2\), the next key step involves simplifying by calculating each term: \((x^2)^2\) becomes \(x^4\), \(2(x^2)\) becomes \(2x^2\), and \(1^2\) results in \(1\). This simplification process involves resolving the powers and coefficients, resulting finally in a cleaner, more manageable expression: \(x^4 + 2x^2 + 1\).

Combining like terms is a critical part of this step. In our context, we confirmed there were no similar terms to combine further, finalizing the expression. This approach ensures clarity and consistency in solving polynomial equations, aiding in the deeper understanding and application of algebraic principles across various mathematical problems.