Sure! Here are some sample questions that could be included in a unit test on polynomial functions and their graphs. Each question covers key concepts associated with polynomials, their characteristics, and their representations:
Multiple Choice Questions
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The degree of a polynomial is defined as:
- A) The highest exponent of the variable in the polynomial
- B) The total number of terms in the polynomial
- C) The highest coefficient of the polynomial
- D) The sum of the exponents of all terms
Answer: A) The highest exponent of the variable in the polynomial
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Which of the following polynomials is a cubic polynomial?
- A) \(2x^2 + 3x + 5\)
- B) \(x^3 - 4x^2 + x - 1\)
- C) \(5 - 2x\)
- D) \(x + 1\)
Answer: B) \(x^3 - 4x^2 + x - 1\)
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What is the end behavior of the polynomial \(f(x) = -2x^4 + 3x^3 - x + 1\)?
- A) Both ends go up
- B) Both ends go down
- C) Left goes down, right goes up
- D) Left goes up, right goes down
Answer: B) Both ends go down (since the leading coefficient is negative and the degree is even)
Short Answer Questions
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State the leading term of the following polynomial:
\(f(x) = 4x^3 - 2x^2 + 7x - 5\)Answer: \(4x^3\)
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Find the zeros of the polynomial \(f(x) = x^2 - 5x + 6\).
Answer: The zeros are \(x = 2\) and \(x = 3\).
Graphing Questions
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Graph the polynomial function \(f(x) = (x - 2)(x + 1)\). Identify the x-intercepts and the y-intercept.
Answer:
- x-intercepts: \(x = 2\) and \(x = -1\)
- y-intercept: \(f(0) = (0 - 2)(0 + 1) = -2\)
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Describe the symmetry of the polynomial function \(g(x) = x^4 - 3x^2 + 2\).
Answer: The function is an even polynomial, and thus it has y-axis symmetry.
Problem Solving
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Given the polynomial \(h(x) = 2x^3 - 6x^2 + 4x\), factor the polynomial completely.
Answer: \(h(x) = 2x(x^2 - 3x + 2) = 2x(x - 1)(x - 2)\)
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Determine whether the polynomial \(p(x) = x^3 - 4x\) has any local maxima or minima. Find their locations.
Answer:
- First, find the critical points by setting \(p'(x) = 0\).
- \(p'(x) = 3x^2 - 4\)
- \(0 = 3x^2 - 4\)
- Thus, \(x^2 = \frac{4}{3}\), leading to \(x = \pm \frac{2}{\sqrt{3}}\).
- Further investigation using the second derivative test or first derivative test would reveal the nature of these critical points.
Extended Response
- Explain how the Fundamental Theorem of Algebra applies to polynomial functions and provide an example.
Answer: The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has as many roots as its degree, counting multiplicities. For example, a polynomial of degree 3, such as \(f(x) = x^3 + 2x^2 + x + 2\), must have exactly 3 roots in the complex number system, which may include real and/or complex roots.
Feel free to use these questions or modify them to fit your specific curriculum or focus areas!