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Questions and Answers
Given $f(x) = -2x(x-5)^2(2x-1)$, what is the multiplicity of the zero at $x = 5$?
Given $f(x) = -2x(x-5)^2(2x-1)$, what is the multiplicity of the zero at $x = 5$?
- 2 (correct)
- 4
- 1
- 3
What is the degree of the polynomial $g(x) = 2(x-1)^2(x+3)$?
What is the degree of the polynomial $g(x) = 2(x-1)^2(x+3)$?
- 4
- 2
- 3 (correct)
- 1
For the function $f(x) = -2x(x-5)^2(2x-1)$, which statement best describes its end behavior?
For the function $f(x) = -2x(x-5)^2(2x-1)$, which statement best describes its end behavior?
- As $x \to \infty$, $f(x) \to -\infty$ and as $x \to -\infty$, $f(x) \to \infty$. (correct)
- As $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to \infty$.
- As $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$.
- As $x \to \infty$, $f(x) \to -\infty$ and as $x \to -\infty$, $f(x) \to -\infty$
A polynomial $p(x)$ has a graph with x-intercepts at $x = -2$ and $x = 3$. The graph touches the x-axis at $x = -2$ and crosses it at $x = 3$. Which of the following could be the equation for $p(x)$, assuming the leading coefficient is 1?
A polynomial $p(x)$ has a graph with x-intercepts at $x = -2$ and $x = 3$. The graph touches the x-axis at $x = -2$ and crosses it at $x = 3$. Which of the following could be the equation for $p(x)$, assuming the leading coefficient is 1?
For the polynomial $g(x) = 2x^3 - 8x^2 + 2x + 12$, what does $g(-1)$ represent?
For the polynomial $g(x) = 2x^3 - 8x^2 + 2x + 12$, what does $g(-1)$ represent?
Given the polynomial $g(x) = 2x^3 - 8x^2 + 2x + 12$, if one of the factors is $(x-3)$, what are the other roots of $g(x)$?
Given the polynomial $g(x) = 2x^3 - 8x^2 + 2x + 12$, if one of the factors is $(x-3)$, what are the other roots of $g(x)$?
Which mathematical notation accurately describes the end behavior of the polynomial $g(x) = -x^3 + 5x^2 - 2x + 1$?
Which mathematical notation accurately describes the end behavior of the polynomial $g(x) = -x^3 + 5x^2 - 2x + 1$?
If a fourth-degree polynomial $T(e)$ models temperature as a function of impulse energy, and it has a complex zero at $3 + 2i$, what must also be a zero of the polynomial?
If a fourth-degree polynomial $T(e)$ models temperature as a function of impulse energy, and it has a complex zero at $3 + 2i$, what must also be a zero of the polynomial?
Given the temperature function $T(e)$ has zeros 5, -2, and a complex zero $3 + 2i$, write $T(e)$ in factored form, not forgetting the leading coefficient $a$.
Given the temperature function $T(e)$ has zeros 5, -2, and a complex zero $3 + 2i$, write $T(e)$ in factored form, not forgetting the leading coefficient $a$.
A polynomial $T(e)$ models temperature (T) based on impulse energy (e). If the y-intercept of the graph is 130, what does this intercept typically represent in the context of this model?
A polynomial $T(e)$ models temperature (T) based on impulse energy (e). If the y-intercept of the graph is 130, what does this intercept typically represent in the context of this model?
Solve the inequality $(x - 3)(x^2 - 25) < 0$. Which of the following intervals represents the solution set?
Solve the inequality $(x - 3)(x^2 - 25) < 0$. Which of the following intervals represents the solution set?
Given $g(x) = 2(x-1)^2(x+3)$, does the graph cross or touch the x-axis at $x=1$?
Given $g(x) = 2(x-1)^2(x+3)$, does the graph cross or touch the x-axis at $x=1$?
Imagine a polynomial function where one of its factors is $(x+a)^n$. Which statement is true of the graph at $x = -a$ if n is odd?
Imagine a polynomial function where one of its factors is $(x+a)^n$. Which statement is true of the graph at $x = -a$ if n is odd?
Consider the function $p(x) = a(x-h)^2+k$. How does changing the value of 'a' affect the graph of the function?
Consider the function $p(x) = a(x-h)^2+k$. How does changing the value of 'a' affect the graph of the function?
What is the y-intercept of $f(x) = -2x(x-5)^2(2x-1)$?
What is the y-intercept of $f(x) = -2x(x-5)^2(2x-1)$?
How does the factored form of a polynomial help in sketching its graph?
How does the factored form of a polynomial help in sketching its graph?
What is the solution to $|x-a|<0$?
What is the solution to $|x-a|<0$?
How many turning points does the polynomial $g(x) = 2(x-1)^2(x+3)$ have?
How many turning points does the polynomial $g(x) = 2(x-1)^2(x+3)$ have?
Given an inequality of the form $f(x) < 0$, what does the solution set represent graphically?
Given an inequality of the form $f(x) < 0$, what does the solution set represent graphically?
Flashcards
What are zeros of a function?
What are zeros of a function?
The values of x where f(x) = 0. They are where the graph intersects or touches the x-axis.
What is multiplicity of a zero?
What is multiplicity of a zero?
The number of times a zero appears as a root of a polynomial. It affects the behavior of the graph at that x-intercept.
What is the degree of a polynomial?
What is the degree of a polynomial?
The highest power of x in the polynomial. It indicates the maximum number of roots and influences the end behavior.
What is End Behavior?
What is End Behavior?
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How do you find zeros from factored form?
How do you find zeros from factored form?
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What is the y-intercept?
What is the y-intercept?
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What is the interval notation?
What is the interval notation?
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What is the bracket notation?
What is the bracket notation?
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Study Notes
- These notes cover polynomial functions, their properties, and related problem-solving techniques.
Problem 1: Analyzing f(x) = -2x(x - 5)²(2x - 1)
- Zeros and Multiplicities:
- x = 0 with multiplicity 1
- x = 5 with multiplicity 2
- x = 1/2 with multiplicity 1
- The degree of the polynomial is 4.
- Leading coefficient is -4
- End Behavior:
- As x approaches negative infinity, f(x) approaches negative infinity.
- As x approaches positive infinity, f(x) approaches negative infinity.
Problem 2: Analyzing g(x) = 2(x - 1)²(x + 3)
- Zeros and Multiplicities:
- x = 1 with multiplicity 2
- x = -3 with multiplicity 1
- The degree of the polynomial is 3.
- The leading coefficient is 2.
- End Behavior:
- As x approaches negative infinity, g(x) approaches negative infinity.
- As x approaches positive infinity, g(x) approaches positive infinity.
Problem 3: Finding the Equation of a Polynomial p(x)
- Use the graph and the given point p(1) = -10 to determine the equation.
- Identify zeros and their multiplicities from the graph.
Problem 4: Analyzing g(x) = 2x³ - 8x² + 2x + 12
- Find g(-1) = -20 and g(0) = 12 which are important for understanding the function's value at those points.
- Determine the zeros of g(x) and their multiplicities by factoring or using numerical methods.
- Write g(x) in factored form after finding the zeros.
- Express the end behavior using mathematical symbols.
Problem 5: Temperature Modeling with a Polynomial
- The polynomial is of least degree 4 and has a zero of 3 + 2i.
- Given data points include (1, -96), (5, 0), (0, 130), (-2, 0), (-1, -120).
- Write out the equation in factored form, including the leading coefficient.
- Find the y-intercept and interpret its meaning.
- Find T(2) and interpret its meaning in the context of the problem.
Problem 6: Solving the Inequality (x - 3)(x² - 25) < 0
- Solve the inequality algebraically.
- Write the solution set in interval notation.
- Begin by factoring: (x - 3)(x - 5)(x + 5) < 0
- Identify critical points: x = -5, 3, 5
- Test intervals to determine when the inequality is true: (-∞, -5), (-5, 3), (3, 5), (5, ∞)
- Solution: (-∞, -5) U (3, 5)
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