Certainly! Here’s a sample unit test on polynomials that covers various concepts such as definitions, operations, and properties. You can modify the questions or use them as a template.
Polynomials Unit Test
Name: ______________________ Date: _______________
Instructions: Answer all the questions to the best of your ability. Show all work for full credit.
Section 1: Definitions (10 points)
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Define a polynomial. Provide an example of a polynomial and identify its degree. (5 points)
Answer: _____________________________________________________________
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What is the difference between a monomial, binomial, and trinomial? Provide an example of each. (5 points)
Answer: _____________________________________________________________
Section 2: Polynomial Operations (30 points)
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Add the following polynomials: \( (3x^2 + 5x - 4) + (2x^2 - 3x + 6) \) (5 points)
Answer: _____________________________________________________________
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Subtract the following polynomials: \( (4x^3 - x^2 + 2x - 1) - (2x^3 + 3x^2 - x + 5) \) (5 points)
Answer: _____________________________________________________________
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Multiply the following polynomials: \( (x + 2)(x^2 - 3) \) (5 points)
Answer: _____________________________________________________________
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Factor the polynomial: \( x^2 - 5x + 6 \) (5 points)
Answer: _____________________________________________________________
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Evaluate the polynomial \( P(x) = 2x^3 - 4x^2 + 3 \) when \( x = 2 \). (5 points)
Answer: _____________________________________________________________
Section 3: Polynomial Properties (20 points)
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State and explain the Remainder Theorem. (5 points)
Answer: _____________________________________________________________
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If \( P(x) = x^3 - 6x^2 + 11x - 6 \), use synthetic division to divide \( P(x) \) by \( (x - 2) \) and find the remainder. (5 points)
Answer: _____________________________________________________________
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What is the Fundamental Theorem of Algebra? (5 points)
Answer: _____________________________________________________________
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Find all the roots (real and complex) of the polynomial \( x^2 + 4x + 5 \). (5 points)
Answer: _____________________________________________________________
Total Points: 100
Answer Key (For Teachers)
Section 1:
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A polynomial is an expression consisting of variables and coefficients, with terms combined using addition, subtraction, and multiplication. Example: \( 2x^3 + 3x^2 - x + 5 \) (Degree is 3).
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Monomial: One term (e.g., \( 3x^2 \)). Binomial: Two terms (e.g., \( x + 2 \)). Trinomial: Three terms (e.g., \( x^2 + 5x + 6 \)).
Section 2:
- \( 5x^2 + 2x + 2 \)
- \( 2x^3 - 4x^2 + 3x - 6 \)
- \( x^3 - 3x + 2 \)
- \( (x - 2)(x - 3) \)
- \( 2(2)^3 - 4(2)^2 + 3 = 16 - 16 + 3 = 3 \)
Section 3:
- The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (x - c) \), the remainder is \( f(c) \).
- Remainder is 0 (If \(0\) is the answer, there are no errors; else show synthetic division).
- The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex root.
- The roots are \( -2 + i \) and \( -2 - i \).
Feel free to add additional problems or modify the test to align with your learning objectives!
3 answers