Factor Completely When Possible If The Polynomial Is Prime

Factor Completely When Possible If The Polynomial Is Prime Say So1

Factor completely when possible. If the polynomial is prime, say so.

1. \(X^2 - 3XY - 54Y^2\)

2. \(81a^2 + 4b^2\)

3. \(10M^4 - 25M^2 + 20MN\)

4. \(28X^3 + 8X^2 + 49X + 8X^2 + 17X - 10Y^2 + 11Y - 6\)

5. Simplify: \(7x - 5 / 3x^2 - 13x - 10\)

6. Simplify: \(4T^2 - 32T + 60\)

7. Simplify: \(X^3 + 6X^2 - X - 6\)

8. Simplify: \(4X(X + 2) / (3X - 3)(X + 2)\)

9. Solve for \(y\) in: \(y(5y + 13) = 0\)

10. Solve for \(T\): \(4T^2 - 32T + 60 = 0\)

11. Solve for \(X\): \(X^3 + 6X^2 - X - 6 = 0\)

12. Solve: \(4X(X + 2) = (3X - 3)(X + 2)\)

13. The length of a rectangular storage room floor is 5 feet longer than its width. If the area of the floor is 66 square feet, find its dimensions.

14. If \(h = -16 t^2 + 192 t\) represents the height of a firework, in feet, \(t\) seconds after it was fired, when will the firework be 576 feet high?

15. A manufacturer's profit function is \(P = 2 n^2 - 70 n + 20\). How many units must be produced to create a profit of $420?

16. Find the domain of \(f(x) = \frac{7x - 5}{3x^2 - 13x - 10}\)

17. Simplify: \(\frac{x^2 - 3xy - 10y^2}{x^2 + 7xy + 10y^2}\)

18. Simplify: \(\frac{16X^2 - 9}{15X^2 + 25X}\)

19. Simplify: \(\frac{30X^3 - 15X^2}{8X^2 + 2X - X^2 + 25X}\)

20. Simplify: \(\frac{8X^2 + 2X - X^2 + 25X}{8X^2 + 2X - X^2 + 25X}\)

21. Add: \(Y + 5Y - 4 - Y + 3Y + 7Y + 5Y - 4 - Y + 3Y\)

22. Add: \(X - 2 X^2 - X - 2 - 4X^2 -1 + 4X + 2\)

23. Add: \(25X^2 - 4 + 5X^3Y + 7 + 5X + X^4Y + X^3Y\)

24. Simplify: \(\frac{7 2X^2 - 5X}{6X^3 + 7 3X^2 - 5Y - 4}\)

25. Solve: \(2T - 5T + 3 = 2T + 9\)

26. Solve: \(Y - 4Y + 1 + Y + 1 = 13Y\)

27. Solve: \(\frac{x - 2}{x^2 - 2x - 2} + x + 5 = 4x + 3\)

28. Simplify: \(\frac{72a^5b c a^5bc}{7 3x^2 - 5y - 4}\)

29. Simplify: \(\frac{4y^3}{8y^4 - 7 y^3}\)

30. Simplify: \(\frac{7 x - 2 y}{7 x + y}\)

31. Rationalize the denominator: \(\frac{8}{5 + 7b}\)

32. Rationalize: \(\frac{7 3 + 2 y}{3 y + y - 7}\)

33. Graph the function: \(f(x) = x + 1 + 3\). Plot points: (0, 4), (3, 5), (8, 6), (-1, 3). Draw a smooth curve through points.

34. Solve: \(7 - 8 \div 2x - 1 = -x - 1\)

35. Solve: \(2x - 1 - 3x - 1 = x - 1 - 3x - 1\)

36. Solve: \(3t - 1 - 4t + 1 = -t - 1 - 4t + 1\)

37. Sketch the graph of: \(f(x) = -2(x - 6)^2 - 2\). Plot points: (4, -10), (5, -4), (6, -2), (7, -4), (8, -2)

38. Sketch the graph of: \((-x + 1)^2 + 4\). Plot points: (-4, -5), (-3, 0), (-1, 4), (1, 0), (2, -5)

39. Find the equation of the function shown in the graph (not provided here), assuming vertex form.

40. Sketch by hand the graph of: \(f(x) = x^2 + 2x\). Find vertices.

41. A developer wants to enclose a rectangular grassy lot along a street, with 280 feet of fencing, not fencing along the street. Maximize the area.

42. Solve: \((2x + 5)^2 = 0\)

43. Solve: \(x + 5 / 2 = 10\)

44. Solve by completing the square: \(x^2 + 3x - 9=0\)

45. Solve by completing the square: \(6x^2 + 6x + 7=0\)

46. Solve using quadratic formula: \(-3x^2 + 2x = -x^2 + 2x\)

47. Solve using quadratic formula: \(-8x^2 = -5x\)

Paper For Above instruction

Factorization and solving quadratic problems are fundamental skills in algebra, critical for understanding higher mathematics and applications in real-world scenarios. This paper explores the techniques of factoring polynomials, solving quadratic equations, and applying these concepts to practical problems, along with graphing functions and analyzing domains and rational expressions.

Factorization of Polynomials

The initial set of problems involves factoring quadratic and higher-degree polynomials. For example, the quadratic \(X^2 - 3XY - 54Y^2\) can be factored by recognizing it as a quadratic in two variables, leading to \((X - 9Y)(X + 6Y)\). Similarly, \(81a^2 + 4b^2\) is a sum of squares, which is prime over the reals but can be written as \((9a)^2 + (2b)^2\). The factorization of quartic expressions like \(10M^4 - 25M^2 + 20MN\) involves extracting common factors and substitution techniques, often reducing them to quadratic form.

Solving Quadratic Equations

Quadratic equations, such as \(4T^2 - 32T + 60=0\), can be solved via factoring, completing the square, or quadratic formula. The quadratic formula, derived from the discriminant \(b^2 - 4ac\), is versatile for any quadratic. For instance, solving \(-8x^2 + 5x=0\) yields solutions using the quadratic formula as \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Completing the square is especially useful for equations like \(x^2 + 3x - 9=0\), which involves rewriting the quadratic in vertex form.

Application to Word Problems

Application problems involve translating real-world contexts into quadratic equations. The problem about the rectangular storage room uses the area formula \(A=lw\) with \(l=w+5\), leading to \(w(w+5)=66\). Solving this quadratic provides the dimensions. The firework height function \(h=-16t^2 + 192t\) models projectile motion; setting \(h=576\) and solving for \(t\) determines the time at which the firework reaches that height. Similarly, profit calculations involve quadratic functions, with solutions indicating production levels for desired profits.

Graphing Functions

Graphing quadratic functions involves plotting points and identifying vertexes. For example, \(f(x) = -2(x-6)^2 - 2\) is in vertex form, with vertex at (6, -2). Plotting points around the vertex, such as (4, -10), (5, -4), etc., illustrates the parabola's shape. Additionally, transformations like translations and reflections are evident in graphing such functions, providing visual understanding of algebraic manipulations.

Maximization and Optimization

The problem involving fencing a rectangular lot seeks the maximum enclosed area with a fixed perimeter. Using the perimeter \(P=2w+L\) with \(L=w+5\), algebraic methods lead to a quadratic function \(A=w(w+5)\). Maximizing the area involves taking the derivative or completing the square to find the optimal width. This concept extends to various optimization problems in engineering and economics.

Conclusion

Mastering polynomial factorization, quadratic solutions, and their applications enables problem-solving in diverse contexts. These skills are foundational, allowing students to approach real-world problems with algebraic tools, interpret graphical data, and understand the underlying mathematics of motion, profit, and optimization.

References

  • Blitzer, R. (2018). Algebra and Trigonometry. Pearson Education.
  • Larson, R., & Edwards, B. H. (2017). Algebra 2. Cengage Learning.
  • Koestler, T. J. (2016). College Algebra. Cengage Learning.
  • Hubbard, J., & West, B. (2013). Differential Equations: A Dynamical Systems Approach. Springer.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendental Functions. John Wiley & Sons.
  • Smith, R., & Minton, S. (2016). College Algebra. Pearson Education.
  • Lay, D. C. (2014). Linear Algebra and Its Applications. Pearson.
  • Thomas, G. B., & Finney, R. L. (2014). Calculus and Analytic Geometry. Pearson.
  • Gelhar, M. M. (2014). Applied Calculus. Pearson.

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