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Chapter 5: Problem 88

Perform the indicated operations \(\left[x^{2}-(5 x-2)\right]\left[x^{2}+(5 x-2)\right]\)

Short Answer

Expert verified
The final expression is \[ x^{4} - 25x^{2} + 20x - 4 \]

Step by step solution

01

- Recognize the pattern

The expression is in the form \[ (a - b)(a + b) \]. The difference of squares formula applies here: \[ (a - b)(a + b) = a^{2} - b^{2} \]. Identify \( a \) and \( b \). In this case, \( a = x^{2} \) and \( b = (5x - 2) \).
02

- Apply the difference of squares formula

Substitute \( a \) and \( b \) into the difference of squares formula: \[ a^{2} - b^{2} = (x^{2})^{2} - (5x - 2)^{2} \]
03

- Simplify the squares

First, calculate \((x^{2})^{2}\): \[ (x^{2})^{2} = x^{4} \]. Next, calculate \((5x - 2)^{2}\): \[ (5x - 2)^{2} = (5x)^{2} - 2 \times 5x \times 2 + 2^{2} = 25x^{2} - 20x + 4 \]
04

- Combine and simplify

Combine the results from Step 3 into the difference of squares formula: \[ x^{4} - (25x^{2} - 20x + 4) \]. Distribute the negative sign through the parentheses: \[ x^{4} - 25x^{2} + 20x - 4 \]

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Key Concepts

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

difference of squares
The difference of squares is a handy formula in algebra. It helps simplify expressions like \((a - b)(a + b)\). This formula states that \((a - b)(a + b) = a^{2} - b^{2}\). The pattern deals with two binomials that are identical except for one having a minus and the other a plus. For instance, in our exercise, \((x^{2} - (5x - 2))(x^{2} + (5x - 2))\), this pattern is evident. Here, \(a = x^{2}\) and \(b = 5x - 2\)\. This makes it easier to simplify by applying the formula directly.
polynomial multiplication
Polynomial multiplication involves multiplying each term in one polynomial by every term in another polynomial. For instance, when working with \( (5x - 2)^{2}\), you multiply every combination of their terms. For better understanding: \(5x \times 5x = 25x^2\), \(5x \times -2 = -10x\), \(-2 \times 5x = -10x\), and finally, \(-2 \times -2 = 4\). When we combine and simplify, \((5x - 2)^{2} = 25x^{2} - 20x + 4\). So whenever multiplying polynomials, carefully expand each term and then combine like terms to simplify.
simplifying expressions
Simplifying expressions means making them easier to understand or solve. After using the difference of squares, we had an expression: \(x^{4} - (25x^{2} - 20x + 4)\). Distributing the negative sign results in \(x^{4} - 25x^{2} + 20x - 4\). Notice how each part undergoes distribution. Simplifying these forms allows figuring out complex equations more straightforwardly. The goal is to end up with the most reduced version of the expression.
binomial expansion
Binomial expansion involves expanding expressions raised to powers, such as \((a + b)^{2}\). For example, \((5x - 2)^{2} \) expands as \((5x - 2)(5x - 2)\). To expand, multiply each term in the first binomial with each term in the second binomial. This results in \((5x)^{2} - 2 \times 5x \times 2 + (2)^{2} = 25x^{2} - 20x + 4\). This is a key concept in resolving the algebra problem and gets used frequently in more advanced mathematics.

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