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Answer :
Let's break down the process of factoring and analyzing the given functions step-by-step.
### Function 1: [tex]\( f(x) = 3x^5 - 16x^4 + 30x^3 - 14x^2 - 13x + 10 \)[/tex]
Factoring:
The polynomial [tex]\( f(x) \)[/tex] factors as:
[tex]\[
f(x) = (x - 1)^2 (3x + 2) (x^2 - 4x + 5)
\][/tex]
Root and zero analysis:
- The factor [tex]\((x - 1)^2\)[/tex] indicates there is a root at [tex]\( x = 1 \)[/tex] with multiplicity 2.
- The linear factor [tex]\((3x + 2)\)[/tex] indicates another root at [tex]\( x = -\frac{2}{3} \)[/tex].
- The quadratic factor [tex]\((x^2 - 4x + 5)\)[/tex] does not factor further over the real numbers, indicating complex roots.
Graph features:
- Intercepts: The x-intercepts are found by setting each factor in [tex]\( f(x) = 0 \)[/tex]. Thus, the intercepts are at [tex]\( x = 1 \)[/tex] and [tex]\( x = -\frac{2}{3} \)[/tex]. There is also a y-intercept at the constant term, which is [tex]\( y = 10 \)[/tex].
### Function 2: [tex]\( g(x) = \frac{5x^3 - 35x^2 + 75x - 45}{x^2 - 1} \)[/tex]
Factoring:
- The numerator factors as: [tex]\( 5(x - 3)^2 \)[/tex].
- The denominator factors as: [tex]\( (x + 1)(x - 1) \)[/tex].
Thus, the function [tex]\( g(x) \)[/tex] can be rewritten as:
[tex]\[
g(x) = \frac{5(x - 3)^2}{(x + 1)(x - 1)}
\][/tex]
Root, asymptote, and hole analysis:
- Numerator zeros (roots of the function): The numerator has a root at [tex]\( x = 3 \)[/tex] with multiplicity 2. This indicates the graph will touch the x-axis and turn around at this point.
- Denominator zeros (asymptotes): The function has vertical asymptotes at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex] because these values make the denominator zero. These result in vertical asymptotes on the graph.
- Holes: Since the numerator and denominator do not share any common factors that cancel out, there are no holes in the graph of [tex]\( g(x) \)[/tex].
Graph features:
- Intercepts and behavior: The x-intercept is [tex]\( x = 3 \)[/tex]. There is no y-intercept because the denominator becomes zero at [tex]\( x = 0 \)[/tex], leading to an undefined value.
### Conclusion
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have been factored and analyzed for intercepts, roots, asymptotes, and holes. Describing the behavior of these functions and highlighting important features can help you sketch their graphs with intercepts and asymptotes, resulting in qualitatively correct shapes based on this analysis.
### Function 1: [tex]\( f(x) = 3x^5 - 16x^4 + 30x^3 - 14x^2 - 13x + 10 \)[/tex]
Factoring:
The polynomial [tex]\( f(x) \)[/tex] factors as:
[tex]\[
f(x) = (x - 1)^2 (3x + 2) (x^2 - 4x + 5)
\][/tex]
Root and zero analysis:
- The factor [tex]\((x - 1)^2\)[/tex] indicates there is a root at [tex]\( x = 1 \)[/tex] with multiplicity 2.
- The linear factor [tex]\((3x + 2)\)[/tex] indicates another root at [tex]\( x = -\frac{2}{3} \)[/tex].
- The quadratic factor [tex]\((x^2 - 4x + 5)\)[/tex] does not factor further over the real numbers, indicating complex roots.
Graph features:
- Intercepts: The x-intercepts are found by setting each factor in [tex]\( f(x) = 0 \)[/tex]. Thus, the intercepts are at [tex]\( x = 1 \)[/tex] and [tex]\( x = -\frac{2}{3} \)[/tex]. There is also a y-intercept at the constant term, which is [tex]\( y = 10 \)[/tex].
### Function 2: [tex]\( g(x) = \frac{5x^3 - 35x^2 + 75x - 45}{x^2 - 1} \)[/tex]
Factoring:
- The numerator factors as: [tex]\( 5(x - 3)^2 \)[/tex].
- The denominator factors as: [tex]\( (x + 1)(x - 1) \)[/tex].
Thus, the function [tex]\( g(x) \)[/tex] can be rewritten as:
[tex]\[
g(x) = \frac{5(x - 3)^2}{(x + 1)(x - 1)}
\][/tex]
Root, asymptote, and hole analysis:
- Numerator zeros (roots of the function): The numerator has a root at [tex]\( x = 3 \)[/tex] with multiplicity 2. This indicates the graph will touch the x-axis and turn around at this point.
- Denominator zeros (asymptotes): The function has vertical asymptotes at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex] because these values make the denominator zero. These result in vertical asymptotes on the graph.
- Holes: Since the numerator and denominator do not share any common factors that cancel out, there are no holes in the graph of [tex]\( g(x) \)[/tex].
Graph features:
- Intercepts and behavior: The x-intercept is [tex]\( x = 3 \)[/tex]. There is no y-intercept because the denominator becomes zero at [tex]\( x = 0 \)[/tex], leading to an undefined value.
### Conclusion
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have been factored and analyzed for intercepts, roots, asymptotes, and holes. Describing the behavior of these functions and highlighting important features can help you sketch their graphs with intercepts and asymptotes, resulting in qualitatively correct shapes based on this analysis.
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