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Chapter 0: Problem 16

Factor the polynomials. $$x^{2}-1$$

Short Answer

Expert verified
The factorization of \(x^2 - 1\) is \((x + 1)(x - 1)\).

Step by step solution

01

Identify the Polynomial

The given polynomial is \(x^{2} - 1\). Notice that it is a quadratic expression.
02

Recognize the Difference of Squares

Observe that the expression \(x^{2} - 1\) fits the form of a difference of squares \(a^2 - b^2\), where \(a = x\) and \(b = 1\).
03

Apply the Difference of Squares Formula

Use the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\). Substituting \(a = x\) and \(b = 1\), we get \(x^{2} - 1 = (x + 1)(x - 1)\).

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Key Concepts

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

Quadratic Expressions
A quadratic expression is a polynomial of degree 2. Examples of quadratic expressions include terms like \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
A key characteristic of quadratic expressions is the highest power of the variable being 2.

In our example, the given polynomial \(x^2 - 1\) is a quadratic expression since it includes \(x^2\) as its highest power.
Difference of Squares
The difference of squares is a specific type of quadratic expression. It takes the form \(a^2 - b^2\). This can be factored into two binomials: \((a + b)(a - b)\).
Let's break it down:
  • \(a^2\): Represents a squared term.
  • \(b^2\): Represents another squared term.
  • \(-\): Indicates a subtraction (difference) between the two squared terms.

In the given exercise, we have \(x^2 - 1\). Here, \(a = x\) and \(b = 1\). Substituting into the difference of squares formula, it simplifies to \((x + 1)(x - 1)\).
Polynomial Factoring
Polynomial factoring is the process of breaking down a polynomial into simpler components, called factors, that when multiplied together give the original polynomial.

There are various methods to factor polynomials, and recognizing specific forms like the difference of squares makes the process easier.
Steps involved generally include:
  • Identifying the polynomial.
  • Recognizing patterns such as the difference of squares or perfect square trinomials.
  • Applying appropriate factoring formulas or techniques.

In our example, we identified \(x^2 - 1\) as a difference of squares and applied the formula \(a^2 - b^2 = (a + b)(a - b)\), resulting in \((x + 1)(x - 1)\).

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Most popular questions from this chapter

After \(t\) hours of operation, an assembly line has assembled \(A(t)=20 t-\frac{1}{2} t^{2}\) power lawn mowers, \(0 \leq t \leq 10 .\) Suppose that the factory's cost of manufacturing \(x\) units is \(C(x)\) dollars, where \(\bar{C}(x)=3000+80 x.\) (a) Express the factory's cost as a (composite) function of the number of hours of operation of the assembly line. (b) What is the cost of the first 2 hours of operation?

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{2}}{x^{5} y}$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$[f(x)]^{3} g(x)$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\sqrt[3]{x} \cdot \sqrt[3]{x^{2}}$$

If \(g(t)=t^{3}+5,\) find \(\frac{g(t+h)-g(t)}{h}\) and simplify.
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