/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Find each product. $$ \left(... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 33

Find each product. $$ \left(2 x^{2}-5\right)\left(2 x^{2}+5\right) $$

Short Answer

Expert verified
The product is \(4x^4 - 25\).

Step by step solution

01

Recognize the Special Product

Identify that the expression \( (2x^2 - 5)(2x^2 + 5) \) is a product of conjugates. The pattern for products of conjugates is \( (a - b)(a + b) = a^2 - b^2 \).
02

Identify a and b

For this product, \( a = 2x^2 \) and \( b = 5 \).
03

Apply the Formula

Substitute \( a \) and \( b \) into the formula \( a^2 - b^2 \). This gives \((2x^2)^2 - 5^2\).
04

Simplify the Squared Terms

Simplify \((2x^2)^2\) to \(4x^4 \), and simplify \(5^2\) to \(25 \). The expression becomes \(4x^4 - 25 \).

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Key Concepts

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

Product of Conjugates
The product of conjugates is an important concept in algebra. Conjugates are pairs of binomials that have the same terms but opposite signs. For example, in the exercise, we have \( 2x^2 - 5 \) and \( 2x^2 + 5 \). Notice the signs. One is a subtraction, and the other is addition. When you multiply conjugates, you get a difference of squares. This special product follows the pattern: \((a - b)(a + b) = a^2 - b^2\). Understanding this pattern helps you solve problems more efficiently.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means breaking down complex expressions into simpler forms. In the given exercise, recognizing that \((2x^2 - 5)(2x^2 + 5)\) forms a special product helps simplify the problem. First, we identify \(a = 2x^2\) and \(b = 5\). Using the product of conjugates formula \((a - b)(a + b) = a^2 - b^2\), we replace \(a\) and \(b\) with their respective values: \((2x^2)^2 - 5^2\). Next, we simplify the squared terms to get \4x^4 - 25\. Simplification helps you solve problems faster and makes complex expressions easy to understand.
Quadratic Expressions
Quadratic expressions are polynomial expressions of degree 2. Typically, they take the form \ax^2 + bx + c\. In our exercise, \(2x^2\) is the quadratic term. When dealing with quadratics in special product forms like the product of conjugates, we can simplify potentially complex polynomials. Here, \4x^4\ from \((2x^2)^2\) is not a simple quadratic but rather a higher-degree polynomial. However, the process of breaking down and simplifying mirrors solving for quadratics. This approach underscores the importance of recognizing patterns and using algebraic rules to simplify and solve expressions effectively.

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Most popular questions from this chapter

Perform each division. $$ \frac{x^{4}-4 x^{3}+5 x^{2}-3 x+2}{x^{2}+3} $$

In Objective \(I,\) we showed how \(6^{\circ}\) acts as 1 when it is applied to the product rule, thus motivating the definition of 0 as an exponent. We can also use the quotient rule to motivate this definition. Because \(25=5^{2},\) the expression \(\frac{25}{25}\) can be written as the quotient of powers of \(5 .\) Write the expression in this way.

To understand how the special product \((a+b)^{2}=a^{2}+2 a b+b^{2}\) can be applied to a purely numerical problem. The number 35 can be written as \(30+5 .\) Therefore, \(35^{2}=(30+5)^{2} .\) Use the special product for squaring a binomial with \(a=30\) and \(b=5\) to write an expression for \((30+5)^{2} .\) Do not simplify at this time.

Perform each indicated operation. Find the difference between the sum of \(5 x^{2}+2 x-3\) and \(x^{2}-8 x+2\) and the sum of \(7 x^{2}-3 x+6\) and \(-x^{2}+4 x-6\)

Perform each division. $$ \frac{3 t^{4}+5 t^{3}-8 t^{2}-13 t+2}{t^{2}-5} $$
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