Math Problem Statement
Math_1319_Quiz 2.pdf
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Solution
Let's go step-by-step to solve each question in your quiz:
1. Evaluate the following:
(a)
Distribute the negative sign:
Now combine like terms:
The result is:
(b)
Recall that powers of follow a cycle:
Since with no remainder, corresponds to .
Thus:
2. Let . Find the solution set (in interval notation) for .
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Identify the roots:
- .
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Analyze the sign changes around these points.
- Between the roots, the factors , , , and may switch sign.
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Since and are always non-negative, the sign changes only depend on and .
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Test intervals between the roots:
- : All factors negative, so .
- : Positive from and , negative from others, so .
- : All factors positive, .
- : Positive except , so .
- : All factors positive, .
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Solution:
on .
3. Solve by factoring.
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Rearrange the equation:
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Factor out :
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Set each factor equal to zero:
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Solution:
Would you like any part of the solutions explained in more detail?
Here are five follow-up questions to explore:
- How can I plot complex numbers on the complex plane?
- What’s the general solution pattern for inequalities involving polynomials?
- How do cycles of apply to more complicated exponents?
- What’s another way to solve quadratic equations like ?
- How do we know when factoring will work versus other methods like the quadratic formula?
Tip: Always analyze roots and their multiplicities when finding where a polynomial is positive or negative. This helps you determine if a root "bounces" or crosses the x-axis.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Polynomial Inequalities
Quadratic Equations
Formulas
Complex number subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Powers of i: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1
Factoring: ax^2 + bx + c = 0
Theorems
Fundamental Theorem of Algebra
Properties of powers of imaginary unit i
Sign analysis for polynomial functions
Suitable Grade Level
Grades 10-12
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