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Chapter 0: Problem 32

Find each product. $$(x+5)(x-5)$$

Short Answer

Expert verified
The product of \((x+5)(x-5)\) is \(x^2 - 25\).

Step by step solution

01

Arrange the terms

Write the product in a clearer form: \((x+5)(x-5) = x(x-5) + 5(x-5)\)
02

Distribute

Now, distribute each term i.e. multiply each term in the first binomial by each term in the second binomial: \( x(x-5) = x^2 - 5x \) and \( 5(x-5) = 5x - 25 \)
03

Simplify

In adding those two, simplify the like terms: \((x^2 - 5x) + (5x - 25) = x^2 - 25\)

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Key Concepts

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

Difference of Squares
The expression \((x+5)(x-5)\) is a classic example of the difference of squares formula. The difference of squares states that when you have a product of two binomials such as \((a+b)(a-b)\), the result is the difference between two squares: \(a^2 - b^2\). This happens because the terms that are linear with respect to \(x\) cancel each other out.
  • In \((x+5)(x-5)\), \(x\) is our \(a\) and \(5\) is our \(b\).
  • Applying the difference of squares formula, you get \(x^2 - 5^2\), which simplifies to \(x^2 - 25\).
This is a quick method and often used to simplify such products without going through detailed multiplication steps.Remember, the formula is valid only when you have a positive and a negative of the same number.
Binomials
Binomials are algebraic expressions containing two terms. For example, \((x+5)\) and \((x-5)\) are both binomials. Each binomial has a form of \(a + b\) or \(a - b\), where \(a\) and \(b\) are individual terms. This gives them flexibility in being part of larger expressions.
  • The structure of a binomial allows you to utilize operations such as distribution.
  • In our exercise, we see how these two binomials are multiplied together to demonstrate the difference of squares.
Recognizing and understanding binomials is crucial because they form the basis of many algebraic operations. Binomials often appear in polynomial equations and are fundamental in expansion, factoring, and solving equations.
Simplification
The final step after working out the arithmetic of binomials involves simplifying the expression. Simplification helps in reducing the expression to its simplest form, making it easier to understand and often easier to use in further calculations.
  • In our step-by-step solution, \(x^2 - 5x + 5x - 25\) simplifies to \(x^2 - 25\).
  • The middle terms, \(-5x\) and \(+5x\), cancel each other out because they are opposites.
This process of simplification not only makes the solution concise but also reveals deeper properties of mathematical expressions like symmetry and hidden factors. Simplification is key to revealing the true form of an expression, which is valuable in solving equations and understanding relationships between terms.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I factored \(4 x^{2}-100\) completely and obtained \((2 x+10)(2 x-10)\).

Exercises \(159-161\) will help you prepare for the material covered in the next section. In parts (a) and (b), complete each statement. a. \(\frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{2}\) b. \(\frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?}\) c. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.

Solve each equation. $$\begin{aligned} 45-[4-2 x-4(x+7)] &=\\\ -4(1+3 x)-[4-3(x+2)-2(2 x-5)] \end{aligned}$$

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.
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