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Let's break down the information given to find the zeros of the function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex] and analyze its end behavior.
### Finding the zeros of the function
To find the zeros of [tex]\( f(x) \)[/tex], we need to solve the equation:
[tex]\[ x^3 + 2x^2 - 5x - 6 = 0 \][/tex]
Numerical solutions of this equation are:
[tex]\[ x_1 \approx -3, \quad x_2 \approx -1, \quad x_3 \approx 2 \][/tex]
These are the points where the function [tex]\( f(x) \)[/tex] crosses the x-axis.
### Analyzing the end behavior
To understand the end behavior of the function, consider the leading term of the polynomial, which is [tex]\( x^3 \)[/tex]. The leading term primarily determines the end behavior of the polynomial function.
For the cubic function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex]:
1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] will be dominated by the leading term [tex]\( x^3 \)[/tex], which also approaches positive infinity because a positive number raised to an odd power (like 3) remains positive and grows very large.
[tex]\[ \lim_{x \to +\infty} f(x) = +\infty \][/tex]
2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] will also be dominated by the leading term [tex]\( x^3 \)[/tex], which approaches negative infinity because a negative number raised to an odd power (like 3) remains negative and grows very large in the negative direction.
[tex]\[ \lim_{x \to -\infty} f(x) = -\infty \][/tex]
### End behavior selection
From the analysis above, we can conclude the following:
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
Therefore, the correct end behavior is:
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
### Final selected points and end behavior
- The zeros of the function [tex]\( f(x) \)[/tex] are approximately at [tex]\( x = -3 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 2 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
### Finding the zeros of the function
To find the zeros of [tex]\( f(x) \)[/tex], we need to solve the equation:
[tex]\[ x^3 + 2x^2 - 5x - 6 = 0 \][/tex]
Numerical solutions of this equation are:
[tex]\[ x_1 \approx -3, \quad x_2 \approx -1, \quad x_3 \approx 2 \][/tex]
These are the points where the function [tex]\( f(x) \)[/tex] crosses the x-axis.
### Analyzing the end behavior
To understand the end behavior of the function, consider the leading term of the polynomial, which is [tex]\( x^3 \)[/tex]. The leading term primarily determines the end behavior of the polynomial function.
For the cubic function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex]:
1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] will be dominated by the leading term [tex]\( x^3 \)[/tex], which also approaches positive infinity because a positive number raised to an odd power (like 3) remains positive and grows very large.
[tex]\[ \lim_{x \to +\infty} f(x) = +\infty \][/tex]
2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] will also be dominated by the leading term [tex]\( x^3 \)[/tex], which approaches negative infinity because a negative number raised to an odd power (like 3) remains negative and grows very large in the negative direction.
[tex]\[ \lim_{x \to -\infty} f(x) = -\infty \][/tex]
### End behavior selection
From the analysis above, we can conclude the following:
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
Therefore, the correct end behavior is:
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
### Final selected points and end behavior
- The zeros of the function [tex]\( f(x) \)[/tex] are approximately at [tex]\( x = -3 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 2 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
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