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Chapter 1: Problem 3

\(y=\left(x^{2}-4\right)\left(9-x^{2}\right)\)

Short Answer

Expert verified
The roots of the function \(y=\left(x^{2}-4\right)\left(9-x^{2}\right)\) are \(x = 2, -2, 3, -3\).

Step by step solution

01

Set Function Equal to Zero

To find the roots of the function, you should first set the function equal to zero as follows: \(0=\left(x^{2}-4\right)\left(9-x^{2}\right)\)
02

Apply Zero Product Property

Next, you should apply zero property rule which states that a product of factors is zero if and only if at least one of the factors is zero: \(x^{2}-4 = 0\) or \(9-x^{2} = 0\)
03

Solve for x

Finally, solve each equation for x. When \(x^{2}-4 = 0\), we get \(x = 2, -2\), and when \(9-x^{2} = 0\), we get \(x = 3, -3\).

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Key Concepts

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

Zero Product Property
The Zero Product Property is a fundamental tool in algebra, especially useful when dealing with polynomial equations. This property states that if the product of two or more factors is zero, at least one of the factors must be zero.
In mathematical terms, if \( a \times b = 0 \), then either \( a = 0 \), \( b = 0 \), or both.
This property is immensely useful because it simplifies the process of finding the roots of polynomial equations.

  • When a polynomial is expressed as a product of factors, you can use this property by setting each factor equal to zero.
  • Each solution from these simpler equations is a potential root of the original polynomial.
The Zero Product Property forms the backbone of understanding how to solve equations involving polynomials by breaking them down into manageable parts.
Solving Quadratics
Quadratic equations are polynomial equations of degree two, generally written in the form \( ax^2 + bx + c = 0 \).
A few methods can be used to solve these equations, among which factoring and the quadratic formula are the most common.
In our exercise, solving the equation by factoring has been utilized.

  • First, rewrite the quadratic expression as a product of simpler polynomial factors.
  • Then, apply the Zero Product Property to find the roots by setting each factor equal to zero.
In the context of this polynomial \( (x^2-4)(9-x^2) = 0 \), both sections are quadratic equations. Solving these simplified quadratic equations for \( x^2 - 4 = 0 \) gives roots \( x = 2 \) and \( x = -2 \),
and for \( 9-x^2 = 0 \) results in \( x = 3 \) and \( x = -3 \).
Solving quadratics efficiently helps in gaining deeper insights into polynomial behaviors.
Factoring Polynomials
Factoring Polynomials involves expressing a polynomial as a product of its simplest polynomial factors. It is an essential skill in simplifying and solving polynomial equations.
When a polynomial is transformed into a factored form, it becomes easier to apply the Zero Product Property.

  • Look for patterns such as the difference of squares, perfect square trinomials, or common factors.
  • The exercise \( (x^2-4)(9-x^2) \) illustrates factoring by recognizing each polynomial as a difference of squares:
  • The expression \( x^2 - 4 \) is the same as \((x - 2)(x + 2)\).
  • Similarly, \( 9 - x^2 \) becomes \((3 - x)(3 + x)\).
This step greatly simplifies the process of solving the equation, as it allows for a straightforward application of the Zero Product Property to find the roots.
Understanding how to factor polynomials not only aids in solving equations but also offers a clearer view of polynomial expressions and their properties.

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Most popular questions from this chapter

Let \(R\) be the region enclosed by the graphs of \(y=2 \ln x\) and \(y=\frac{x}{2},\) and the lines \(x=2\) and \(x=8\). (a) Find the area of \(R\) . (b) Set up, but do not integrate, an integral expression, in terms of a single variable, for the volume of the solid generated when \(R\) is revolved about the \(x\)-axis. (c) Set up, but do not integrate, an integral expression, in terms of a single variable, for the volume of the solid generated when R is revolved about the line \(x=-1\)

Now evaluate the following integrals. \(\int \frac{\cos \left(\frac{3}{x}\right)}{x^{2}} d x\)

If \(f(x)=\cos ^{2} x,\) then \(f^{\prime}(\pi)=\) (A) -2 (B) 0 (C) 1 (D) 2

Evaluate the following integrals. \(\int \frac{d x}{\sqrt{x}(1+2 \sqrt{x})}\)

If \(\frac{d y}{d x}=\frac{\sin x}{\cos y}\) and \(y(0)=\frac{3 \pi}{2},\) find an equation for \(y\) in terms of \(x\)
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