Form A Polynomial Whose Real Zeros And Degree Are Given

Form A Polynomial Whose Real Zeros And Degree Are Givenzerosminu

Form a polynomial whose real zeros and degree are given. ​Zeros: minus−44​, ​0, 11​;   ​​ degree: 3. Type a polynomial with integer coefficients and a leading coefficient of 1. ​f(x)equals= ​(Simplify your​ answer.) 2) Form a polynomial whose zeros and degree are given. ​Zeros: negative 3−3​, multiplicity​ 1;    negative 4−4​, multiplicity​ 2; degree 3. Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. ​f(x)equals=nothing ​(Simplify your​ answer.) For the polynomial function​ below: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the​ x-axis at each​ x-intercept. (c) Determine the behavior of the graph near each​ x-intercepts. (d) Determine the maximum number of turning points on the graph. (e) Determine the end​ behavior; that​ is, find the power function that the graph of f resembles for large values of |x| ​ f(x) = - 7(x – 4)(x + 5)^2 ​(Type each answer only once. Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.) For the polynomial function​ below: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the​ x-axis at each​ x-intercept. (c) Determine the behavior of the graph near each​x-intercept. †(d) Determine the maximum number of turning points on the graph. †(e)Determine the end​behavior, that​ is, find the power function that the graph of f resembles for large values of |x| f(x) = -6(x^2 + 4)(x – 7)^3 ​(Type each answer only once. Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)

Paper For Above instruction

The process of forming polynomials based on given zeros and degrees involves applying fundamental algebraic principles to ensure the polynomials accurately reflect the specified roots and multiplicities. This paper explores the systematic method to construct such polynomials, details the behavior of their graphs, and provides insights into their end behaviors and maximum turning points, exemplified by specific cases.

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients, structured to incorporate roots (or zeros) with specified multiplicities and degrees. Creating a polynomial from the given roots involves utilizing the fact that if a root r has multiplicity m, then (x - r) appears m times as a factor. Understanding the zeros, their multiplicities, and how the polynomial's degree constrains its shape and behavior is critical in graphing and analyzing polynomial functions. This paper aims to demonstrate the process of constructing such polynomials and analyzing their properties based on the provided zeros with multiplicities and degrees.

Constructing Polynomials from Zeros and Degrees

The first task involves creating a polynomial with specified zeros: -44, 0, and 11, and degree 3. Since the degree is 3 and there are three zeros, each zero must have multiplicity 1 to satisfy the degree condition. The polynomial, therefore, is expressed as:

f(x) = (x + 44)(x)(x - 11)

Expanding this product yields:

f(x) = x(x + 44)(x - 11) = x(x^2 + 33x - 484) = x^3 + 33x^2 - 484x

This polynomial has degree 3 with integer coefficients and a leading coefficient of 1. The zeros are -44, 0, and 11, matching the prescribed zeros and degree.

Second Polynomial Construction with Given Zeros and Multiplicities

The second polynomial involves zeros at -3 (multiplicity 1) and -4 (multiplicity 2), with total degree 3. Constructing the factors:

f(x) = (x + 3)(x + 4)^2

Expanding the quadratic factor:

(x + 4)^2 = x^2 + 8x + 16

Multiplying through:

f(x) = (x + 3)(x^2 + 8x + 16) = x^3 + 8x^2 + 16x + 3x^2 + 24x + 48 = x^3 + 11x^2 + 40x + 48

This polynomial has degree 3, integer coefficients, and a leading coefficient of 1, as specified. Its zeros are at -3 and -4, with -4 having multiplicity 2.

Analysis of the Polynomial Function f(x) = -7(x – 4)(x + 5)^2

a) Zeros and Multiplicities

The zeros are located at x = 4 and x = -5. The zero at x = 4 has multiplicity 1, and at x = -5, multiplicity 2.

b) Cross or Touch the x-axis

The graph crosses the x-axis at x = 4 because of odd multiplicity (1). It touches the x-axis at x = -5 because of even multiplicity (2).

c) Behavior Near Zeros

Near x = 4, the graph crosses the x-axis, moving from positive to negative or vice versa. At x = -5, the graph touches the x-axis and turns around, not crossing it, indicating a local extremum at that point.

d) Maximum Number of Turning Points

The degree of the polynomial is five (obtained by multiplying the factors: degree 1 at x=4 and degree 2 at x=-5). The maximum number of turning points for a degree n polynomial is n - 1. Therefore, the maximum number of turning points here is 4.

e) End Behavior

As |x| approaches infinity, the dominant term of the polynomial is -7x^3 (since the multiplication yields degree 3). The leading coefficient is negative, so the behavior is:

- As x approaches +∞, f(x) approaches -∞.

- As x approaches -∞, f(x) approaches +∞.

This reflects the end behavior of a degree 3 polynomial with negative leading coefficient.

Analysis of the Polynomial Function f(x) = -6(x^2 + 4)(x – 7)^3

a) Zeros and Multiplicities

Zeros are at x = -2 (from the quadratic factor x^2 + 4, which has zeros at x = ±2i, but these are complex and non-real zeros) and at x = 7 (from the factor (x - 7)^3). Since only real zeros are considered, x = 7 is the relevant zero with multiplicity 3.

b) Cross or Touch the x-axis

The polynomial touches the x-axis at x = 7 because of odd multiplicity (3), so the graph crosses the x-axis at x=7.

c) Behavior Near Zeros

At x=7, due to odd multiplicity, the graph passes through the x-axis, changing sign. The zeros from x^2 + 4 are complex, so no real x-intercepts arise from them, and the graph does not touch or cross the x-axis at those points.

d) Maximum Number of Turning Points

The degree of this polynomial is the sum of the degrees of its factors: (x^2 + 4) is degree 2, and (x - 7)^3 is degree 3, so total degree = 2 + 3 = 5. The maximum number of turning points is degree minus 1, i.e., 4.

e) End Behavior

The leading term comes from (x^2)(x^3) multiplied by -6, which simplifies to -6x^5. Since the leading coefficient is negative, as x approaches +∞, f(x) approaches -∞; as x approaches -∞, f(x) approaches +∞, reflecting typical behavior of odd-degree polynomials with negative leading coefficients.

Conclusion

The construction and analysis of these polynomials demonstrate key principles in polynomial algebra and graph behavior. Building from zeros and their multiplicities guides the form of the polynomial, while understanding degree and leading coefficient predicts the end behavior and the maximum number of turning points. These insights are crucial for graphing and analyzing polynomial functions in advanced mathematics and calculus.

References

  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals (11th ed.). John Wiley & Sons.
  • Swokowski, E. W., & Cole, J. A. (1998). Algebra and Trigonometry with Analytic Geometry. PWS Publishing.
  • Larson, R., & Edwards, B. H. (2017). Calculus (11th ed.). Cengage Learning.
  • Fitzpatrick, P. (2013). Algebra and Trigonometry: Functions and Applications. McGraw-Hill Education.
  • Ratti, R. L., & McWaters, M. (2008). Precalculus: Mathematics for Calculus (4th ed.). Pearson.
  • Houghton Mifflin Harcourt. (2010). Precalculus (Teacher Edition). Houghton Mifflin Harcourt.
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry. Pearson.
  • Zitarelli, D. (2012). The Art and Science of Mathematics. Springer.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Brooks Cole.
  • Larson, R., & Hostetler, R. P. (2007). Algebra and Trigonometry. Houghton Mifflin.

Related Assignments