Problem 58 Perform the indicated operations... [FREE SOLUTION]

91影视

Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 58

Perform the indicated operations and simplify. $$\left(x+\left(2+x^{2}\right)\right)\left(x-\left(2+x^{2}\right)\right)$$

Short Answer

Expert verified

The simplified expression is $-x^4 - 3x^2 - 4$.

Step by step solution

Understand the Problem

The problem involves performing operations on two binomials: $ \left(x + (2 + x^{2})\right)$ and $ \left(x - (2 + x^{2})\right)$. These binomials are in the form $(a + b)(a - b)$, which is a difference of squares.

Apply the Difference of Squares Formula

The difference of squares formula is $(a + b)(a - b) = a^2 - b^2$. Here, $a = x$ and $b = 2 + x^2$. Use this formula to simplify the expression.

Calculate $a^2$

Square $a$, which is $x$. Thus, $a^2 = x^2$.

Calculate $b^2$

Square $b$, which is $2 + x^2$. This gives $b^2 = (2 + x^2)^2 = 4 + 4x^2 + x^4$.

Perform the Subtraction

Subtract $b^2$ from $a^2$ using the results from Steps 3 and 4: $x^2 - (4 + 4x^2 + x^4)$.

Simplify the Expression

Distribute the negative sign and simplify: $x^2 - 4 - 4x^2 - x^4 $. Combine like terms to get the final expression: $-x^4 - 3x^2 - 4$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Operations

Binomial operations are a key concept in algebra involving expressions with two terms. These operations often include addition, subtraction, multiplication, and division within a polynomial expression. Let鈥檚 dive into how these operations function.

**Understanding Product of Binomials**: When multiplying binomials, you apply the distributive property. In this exercise, we have two binomials: $ (x + (2 + x^2)) $ and $ (x - (2 + x^2)) $. This specific case describes a pair of binomials that form a difference of squares.
**Recognizing Patterns**: The form $(a + b)(a - b)$ simplifies using the difference of squares formula: $a^2 - b^2$. Notice how recognizing such patterns makes operations more efficient.

Whenever you are faced with binomial expressions, look for opportunities to apply known identities and formulas. Identifying these can simplify the process and reduce potential errors.

Squaring Expressions

Squaring expressions means multiplying an expression by itself. This is a common operation in algebra and understanding it is crucial for simplifying polynomial expressions.

**Individual Terms**: Consider an expression $b = 2 + x^2$. Squaring $b$ results in $ (2 + x^2)^2 $.
**Expansion**: To compute $b^2$, expand it: $b^2 = (2 + x^2) \cdot (2 + x^2)$. This requires applying the distributive property, resulting in $4 + 4x^2 + x^4$.

Squaring expressions using these methods helps retain clear oversight on how each term contributes to the final polynomial. Practicing expansion ensures that you become adept at breaking down and simplifying increasingly complex expressions.

Polynomial Simplification

Simplification involves reducing expressions to their simplest form. It鈥檚 like cleaning up your math work, making it neater and easier to interpret.

**Combining Like Terms**: In the operation $x^2 - (4 + 4x^2 + x^4)$, distribute the negative sign across the parentheses to get $x^2 - 4 - 4x^2 - x^4$.
**Simplification Process**: Begin by combining like terms. Here, combine $-4x^2$ with $x^2$ to result in $-3x^2$, and you are left with $-x^4 - 3x^2 - 4$.

Mastering polynomial simplification not just elevates your skill in solving algebra problems but also heightens your ability to recognize efficient shortcuts. Regular practice will enhance your capability to strategize the simplification process intuitively.

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91影视

Perform the indicated operations and simplify. $$\left(x+\left(2+x^{2}\right)\right)\left(x-\left(2+x^{2}\right)\right)$$

Short Answer

Step by step solution

Understand the Problem

Apply the Difference of Squares Formula

Calculate \(a^2\)

Calculate \(b^2\)

Perform the Subtraction

Simplify the Expression

Key Concepts

Binomial Operations

Squaring Expressions

Polynomial Simplification

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