Quiz 4 Week 6 Math 107 College Algebra Instructor Staci Gash

Quiz 4 Week 6math 107 College Algebrainstructor Staci Gashinstructi

This quiz covers material through week 6. Students are instructed to show their work clearly for each problem to receive partial credit, and solutions can be submitted either typed or handwritten (scanned).

The specific problems include simplifying complex expressions, solving quadratic equations using the quadratic formula, analyzing quadratic functions for vertex, axis of symmetry, and maximum or minimum values, performing various algebraic operations, solving absolute value inequalities with interval notation, factoring, simplifying radicals, and solving polynomial equations. The emphasis is on demonstrating understanding of processes and algebraic manipulation as per the topics covered in the course up to week 6.

Paper For Above instruction

This paper provides comprehensive solutions and explanations for the assigned problems from the quiz covering algebraic simplifications, quadratic equations, function analysis, inequalities, radicals, factoring, and polynomial equations. Each problem is approached systematically, highlighting the methods and reasoning used to arrive at the correct solution, in accordance with college-level algebra standards.

Simplification and Operations of Complex Expressions

First, simplifying complex expressions and performing algebraic operations form the foundation of college algebra skills. For instance, given the expression (6 + 5i)^2, we expand using the binomial theorem:

(6 + 5i)^2 = 6^2 + 2 6 5i + (5i)^2 = 36 + 60i + 25i^2.

Since i^2 = -1, we substitute and get 36 + 60i - 25 = (36 - 25) + 60i = 11 + 60i.

Similarly, operations like multiplying binomials or dealing with complex conjugates follow standard algebraic rules.

Solving Quadratic Equations

Quadratic equations such as 3x^2 + x – 2 = 0 are best solved using the quadratic formula:

x = [-b ± √(b^2 – 4ac)] / 2a.

Here, a = 3, b = 1, c = -2, so:

x = [-1 ± √(1^2 - 43(-2))] / (2*3) = [-1 ± √(1 + 24)] / 6 = [-1 ± √25] / 6.

Therefore, x = (-1 ± 5) / 6, leading to solutions x = 4/6 = 2/3 and x = -6/6 = -1.

Analyzing Quadratic Functions

For the quadratic function f(x) = 2x^2 + 6x + 8, we analyze features such as the vertex, axis of symmetry, and extremum (max or min).

The vertex of a parabola given by y = ax^2 + bx + c occurs at x = -b / 2a:

x = -6 / (2*2) = -6 / 4 = -1.5.

Plugging back into f(x):

f(-1.5) = 2(-1.5)^2 + 6(-1.5) + 8 = 2*2.25 - 9 + 8 = 4.5 - 9 + 8 = 3.5.

The parabola opens upward (since a = 2 > 0), indicating a minimum value at the vertex, which is 3.5.

Solving Absolute Value Inequalities

Consider the inequality |x - 7| ≤ 4. Its solution set comprises all x within 4 units of 7, i.e.,

x - 7 ≤ 4 and x - 7 ≥ -4.

Solving both: x ≤ 11 and x ≥ 3.

Thus, the solution in interval notation is [3, 11].

Similarly, for |x + 5| ≥ 3, the solution involves the union: x + 5 ≥ 3 or x + 5 ≤ -3, leading to x ≥ -2 or x ≤ -8, with the union expressed as (-∞, -8] ∪ [-2, ∞).

Factorization and Radical Simplification

Factoring quadratics without solutions, e.g., x^2, is straightforward: x^2 = x * x.

More complex cases like x^2 + 2x + 1 factor as (x + 1)^2, which is a perfect square.

Simplifying radicals such as √(50) involves prime factorization:

√(50) = √(25 * 2) = 5√2.

Polynomial Operations and Equations

To combine polynomials such as (x^2 - 3x + 4) + (2x^2 + x - 5) = 3x^2 - 2x - 1, you combine like terms.

Solve quartic equations or higher-degree polynomials by factoring, synthetic division, or using the quadratic formula on quadratic factors.

Discussion on Critical Concepts and Techniques

Key skills in college algebra include understanding how to manipulate algebraic expressions, solving equations robustly, analyzing functions graphically and algebraically, and applying inequalities to define solution sets accurately. Mastery of these concepts enables students to progress to calculus and other advanced mathematics topics.

Concluding Remarks

The section of the course covering quadratic functions, polynomial operations, radicals, and inequalities forms the core analytical toolkit for understanding algebraic relationships. Applying these methods accurately ensures correct problem-solving and solid mathematical reasoning essential for higher-level mathematics courses.

References

  • Larson, R., Hostetler, R. P., & Edwards, B. H. (2019). Algebra and Trigonometry. Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2016). Algebra: A Combined Approach. Wiley.
  • Blitzer, R. (2014). Algebra and Trigonometry. Pearson.
  • Sullivan, M. (2018). College Algebra (6th ed.). Pearson.
  • Lay, D. C. (2016). Linear Algebra and Its Applications. Pearson.
  • Swokowski, E. W., & Cole, J. A. (2011). Algebra and Trigonometry with Analytic Geometry. Cengage.
  • Larson, R., & Hostetler, R. P. (2018). College Algebra with Corequisite Support. Cengage Learning.
  • Divakaruni, R. K. (2020). Fundamentals of College Algebra. Prentice Hall.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Houghton Mifflin Harcourt. (2018). Common Core Algebra I. Houghton Mifflin Harcourt.

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