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Chapter 4: Problem 105

Find the zeros of the function. $$g(x)=-x^{3}-2 x^{2}[4.1]$$

Short Answer

Expert verified
The zeros are \( x = 0 \) and \( x = -2 \).

Step by step solution

01

Set the Function Equal to Zero

To find the zeros of the function, set the function equal to zero: \( -x^3 - 2x^2 = 0 \)
02

Factor the Function

Factor out the common factor, which is \(-x^2\): \( -x^2(x + 2) = 0 \)
03

Solve Each Factor for Zero

Set each factor equal to zero and solve: \( -x^2 = 0 \) \( x + 2 = 0 \) From \( -x^2 = 0 \), we get \( x = 0 \). From \( x + 2 = 0 \), we get \( x = -2 \).
04

List All Solutions

The zeros of the function are \( x = 0 \) and \( x = -2 \).

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Key Concepts

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

factoring polynomial functions
When working with polynomials, one of the most useful techniques is factoring. This basically means breaking down a polynomial into simpler 'factors' that you can multiply together to get the original polynomial.

Consider the function from our exercise: \( g(x) = -x^3 - 2x^2 \). In order to solve for the zeros, we start by factoring. Factor out the greatest common factor (GCF), which is the largest term that appears in each part of the polynomial. Here, the GCF is \( -x^2 \).

When you factor \( -x^2 \) out of each term, you get:
\( -x^2 (x + 2) = 0 \). Breaking it down into these factors makes it much simpler to solve.
solving polynomial equations
After factoring, the next step is to solve the polynomial equation. The equation \( -x^2(x + 2) = 0 \) becomes more straightforward now. To find the solutions, or roots, you need to set each factor equal to zero and solve:
  • \( -x^2 = 0 \)
  • \( x + 2 = 0 \)

Solving each equation separately: \( -x^2 = 0 \) implies that \( x = 0 \) because zero squared is still zero.
Solving \( x + 2 = 0 \) gives us another root: \( x = -2 \). So, the solutions to the polynomial are 0 and -2.
roots of equations
Roots (or zeros) of a polynomial are the x-values that make the polynomial equal to zero. In our exercise, we found the roots by factoring and solving the polynomial.

For the polynomial \( g(x) = -x^3 - 2x^2 \), the roots were calculated to be \ x = 0 \ and \ x = -2 \.

These roots tell us where the graph of the polynomial function will intersect the x-axis. To summarize, finding the roots involves:
  • Setting the polynomial equation equal to zero
  • Factoring the polynomial
  • Solving the simpler equations derived from the factors

This process helps us fully understand the behavior of the polynomial function.

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Most popular questions from this chapter

What does Descartes' rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? $$f(x)=3 x^{5}-2 x^{2}+x-1$$

Find the nonlinear asymptote of the function. $$f(x)=\frac{x^{4}+3 x^{2}}{x^{2}+1}$$

What does Descartes' rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? $$h(t)=-4 t^{5}-t^{3}+2 t^{2}+1$$

Consider the function $$g(x)=x^{2}+5 x-14$$ Given an output of \(-20,\) find the corresponding inputs.

Use synthetic division and the remainder theorem to find the zeros. $$f(x)=-x^{3}+3 x^{2}+6 x-8$$
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