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Chapter 2: Problem 12

Find all the zeros of the function. \(f(x)=(x+5)(x-8)^{2}\)

Short Answer

Expert verified
The zeros of the function \(f(x)=(x+5)(x-8)^{2}\) are \(x = -5\) and \(x = 8\). \((x-8)^{2}\) indicates a double root at \(x=8\), meaning that the function crosses the x-axis at \(x=-5\) and touches it at \(x=8\).

Step by step solution

01

Set the function equals to zero

First, set the function equal to zero to solve for x-values. Formulate equation:\(f(x) = 0(x+5)(x-8)^{2}=0\)
02

Use the Zero Product Property

According to the zero product property, if the product of multiple factors is equal to zero, then at least one of the factors must equal zero.Hence, we can write the equation as follows: \(x + 5 = 0\) or \((x -8)^2 = 0\)
03

Solve for x

Now, solve each equation for \(x\). So for the equation \(x + 5 = 0\), subtracting 5 from both sides gives:\(x = -5\) For the equation \((x -8)^2 = 0\), applying square root on both sides gives:\(x = 8\)
04

Verifying the zeros

After solving, we find that \(x = -5\) and \(x = 8\) are zeros of the function \(f(x)=(x+5)(x-8)^{2}\), that is, the function equals zero when \(x = -5\) and \(x = 8\). The function has a double root at \(x=8\) corresponding to the \((x 鈭� 8)^2\) term.

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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 crucial concept in algebra which states that if the product of several factors is zero, then at least one of the factors must be zero. This property simplifies the process of finding the zeros of a polynomial function. When you have an equation such as \[(x+5)(x-8)^2 = 0\]the Zero Product Property allows us to break it down:
  • Either \(x + 5 = 0\) or \((x - 8)^2 = 0\)
  • This means solving two simpler equations separately.
  • First, solve \(x + 5 = 0\) which gives \(x = -5\).
  • Next, solve \((x - 8)^2 = 0\), which requires taking the square root.
  • This results in \(x = 8\).
Using the Zero Product Property simplifies complex polynomial equations into solvable parts. It鈥檚 an invaluable tool for quickly identifying the roots of a function.
Double roots
In mathematics, a double root occurs when a particular root appears twice in a polynomial equation. Double roots are a result of terms raised to an even power, like a squared term.In the function \(f(x) = (x+5)(x-8)^2\), the factor \((x-8)\) is squared. This tells us that the root \(x=8\) is a double root. Double roots have specific characteristics:
  • The graph of the polynomial touches the x-axis at the double root but does not cross it.
  • They can also indicate a symmetric or repeating behavior in functions.
  • Substituting the double root back into the original function will confirm that the output is zero, which verifies the root.
Recognizing double roots is important, as they affect the behavior and symmetry of the polynomial's graph. They provide insight into how the function behaves at and around those x-values.
Polynomial equations
Polynomial equations involve expressions that are made up of variables and coefficients, linked together using addition, subtraction, and multiplication, and having non-negative integer exponents. The function presented in this exercise, \(f(x) = (x + 5)(x - 8)^2\), is a polynomial equation.Polynomial equations are essential in mathematics due to their widespread applications:
  • They can have multiple roots, including real and complex solutions.
  • Identifying whether a root is a single or double is key to understanding the polynomial's characteristics.
  • Polynomials can model various real-world situations, making them useful in physics, engineering, and economics.
To efficiently find the roots of polynomial equations, properties like the Zero Product Property, among other algebraic techniques, are used. Comprehending these equations' structure and solutions equips students with valuable problem-solving tools.

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