Patricia Is Studying A Polynomial Function \[$ F(x) \$\]. Three Given Roots Of \[$ F(x) \$\] Are \[$-11-\sqrt{2} I\$\], \[$3+4 I\$\], And \[$10\$\]. Patricia Concludes That \[$ F(x) \$\] Must Be A

by ADMIN 197 views

=====================================================

Introduction

Polynomial functions are a fundamental concept in mathematics, and understanding their properties and behavior is crucial for solving various mathematical problems. In this article, we will explore the concept of polynomial functions and their roots, and discuss how to determine the nature of a polynomial function given its roots.

What are Polynomial Functions?

A polynomial function is a function that can be written in the form:

f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0

where an,an−1,…,a1,a0a_n, a_{n-1}, \ldots, a_1, a_0 are constants, and nn is a non-negative integer. The degree of the polynomial is nn, and the coefficients an,an−1,…,a1,a0a_n, a_{n-1}, \ldots, a_1, a_0 determine the shape of the graph of the function.

What are Roots of a Polynomial Function?

The roots of a polynomial function are the values of xx that make the function equal to zero. In other words, if f(x)=0f(x) = 0, then xx is a root of the function. The roots of a polynomial function can be real or complex numbers.

Given Roots of a Polynomial Function

In this article, we are given three roots of a polynomial function f(x)f(x): −11−2i-11-\sqrt{2} i, 3+4i3+4 i, and 1010. We need to determine the nature of the polynomial function f(x)f(x) given these roots.

Determining the Nature of a Polynomial Function

To determine the nature of a polynomial function given its roots, we can use the following steps:

  1. Write the polynomial function in factored form: We can write the polynomial function in factored form as:

f(x)=a(x−r1)(x−r2)(x−r3)…(x−rn)f(x) = a(x - r_1)(x - r_2)(x - r_3) \ldots (x - r_n)

where r1,r2,…,rnr_1, r_2, \ldots, r_n are the roots of the function. 2. Determine the degree of the polynomial: The degree of the polynomial is equal to the number of roots. 3. Determine the coefficients of the polynomial: We can determine the coefficients of the polynomial by expanding the factored form of the polynomial.

Determining the Coefficients of the Polynomial

To determine the coefficients of the polynomial, we can expand the factored form of the polynomial:

f(x)=a(x−(−11−2i))(x−(3+4i))(x−10)f(x) = a(x - (-11-\sqrt{2} i))(x - (3+4 i))(x - 10)

Expanding the product, we get:

f(x)=a(x2+(11+2i)x+(121+2i+22i2))(x−10)f(x) = a(x^2 + (11+\sqrt{2} i)x + (121+2 i + 2 \sqrt{2} i^2))(x - 10)

Simplifying the expression, we get:

f(x)=a(x3−10x2+(121+2i+22i2)x+1100)f(x) = a(x^3 - 10x^2 + (121+2 i + 2 \sqrt{2} i^2)x + 1100)

Determining the Value of the Coefficient

To determine the value of the coefficient aa, we can use the fact that the polynomial function has a root at x=10x = 10. Substituting x=10x = 10 into the polynomial function, we get:

f(10)=a(103−102(10)+(121+2i+22i2)(10)+1100)f(10) = a(10^3 - 10^2(10) + (121+2 i + 2 \sqrt{2} i^2)(10) + 1100)

Simplifying the expression, we get:

f(10)=a(1000−1000+1210+20i+202i2+1100)f(10) = a(1000 - 1000 + 1210 + 20 i + 20 \sqrt{2} i^2 + 1100)

Simplifying further, we get:

f(10)=a(3210+20i+202i2)f(10) = a(3210 + 20 i + 20 \sqrt{2} i^2)

Since f(10)=0f(10) = 0, we can set the expression equal to zero and solve for aa:

3210+20i+202i2=03210 + 20 i + 20 \sqrt{2} i^2 = 0

Simplifying the expression, we get:

3210+20i−202i=03210 + 20 i - 20 \sqrt{2} i = 0

Simplifying further, we get:

(3210−202)+20i=0(3210 - 20 \sqrt{2}) + 20 i = 0

Simplifying further, we get:

(3210−202)=0(3210 - 20 \sqrt{2}) = 0

Simplifying further, we get:

3210=2023210 = 20 \sqrt{2}

Simplifying further, we get:

160.5=2160.5 = \sqrt{2}

Simplifying further, we get:

160.52=2160.5^2 = 2

Simplifying further, we get:

25804.25=225804.25 = 2

Simplifying further, we get:

12902.125=112902.125 = 1

Simplifying further, we get:

12902.125=112902.125 = 1

This is a contradiction, and therefore, the value of aa is not a real number.

Conclusion

In this article, we have discussed the concept of polynomial functions and their roots, and have determined the nature of a polynomial function given its roots. We have shown that the polynomial function f(x)f(x) has a root at x=10x = 10, and have determined the value of the coefficient aa. However, we have found that the value of aa is not a real number, and therefore, the polynomial function f(x)f(x) is not a real polynomial function.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "Roots of a Polynomial Function" by Wolfram MathWorld
  • [3] "Determining the Nature of a Polynomial Function" by Khan Academy

Further Reading

  • "Polynomial Functions and Their Roots" by MIT OpenCourseWare
  • "Algebraic Geometry" by Springer
  • "Calculus" by Michael Spivak

=====================================================

Introduction

In our previous article, we discussed the concept of polynomial functions and their roots, and determined the nature of a polynomial function given its roots. In this article, we will answer some frequently asked questions about polynomial functions and their roots.

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form:

f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0

where an,an−1,…,a1,a0a_n, a_{n-1}, \ldots, a_1, a_0 are constants, and nn is a non-negative integer.

Q: What are the roots of a polynomial function?

A: The roots of a polynomial function are the values of xx that make the function equal to zero. In other words, if f(x)=0f(x) = 0, then xx is a root of the function.

Q: How do I determine the nature of a polynomial function given its roots?

A: To determine the nature of a polynomial function given its roots, you can use the following steps:

  1. Write the polynomial function in factored form: You can write the polynomial function in factored form as:

f(x)=a(x−r1)(x−r2)(x−r3)…(x−rn)f(x) = a(x - r_1)(x - r_2)(x - r_3) \ldots (x - r_n)

where r1,r2,…,rnr_1, r_2, \ldots, r_n are the roots of the function. 2. Determine the degree of the polynomial: The degree of the polynomial is equal to the number of roots. 3. Determine the coefficients of the polynomial: You can determine the coefficients of the polynomial by expanding the factored form of the polynomial.

Q: How do I determine the coefficients of a polynomial function?

A: To determine the coefficients of a polynomial function, you can expand the factored form of the polynomial. For example, if the polynomial function is:

f(x)=a(x−r1)(x−r2)(x−r3)f(x) = a(x - r_1)(x - r_2)(x - r_3)

you can expand the product to get:

f(x)=a(x3−(r1+r2+r3)x2+(r1r2+r1r3+r2r3)x−r1r2r3)f(x) = a(x^3 - (r_1 + r_2 + r_3)x^2 + (r_1r_2 + r_1r_3 + r_2r_3)x - r_1r_2r_3)

Q: What is the relationship between the roots of a polynomial function and its coefficients?

A: The roots of a polynomial function are related to its coefficients by the following formula:

r1+r2+…+rn=−an−1anr_1 + r_2 + \ldots + r_n = -\frac{a_{n-1}}{a_n}

r1r2+r1r3+…+rn−1rn=an−2anr_1r_2 + r_1r_3 + \ldots + r_{n-1}r_n = \frac{a_{n-2}}{a_n}

…\ldots

r1r2…rn=(−1)na0anr_1r_2 \ldots r_n = (-1)^n\frac{a_0}{a_n}

Q: How do I use the relationship between the roots and coefficients to determine the nature of a polynomial function?

A: To use the relationship between the roots and coefficients to determine the nature of a polynomial function, you can use the following steps:

  1. Determine the roots of the polynomial function: You can determine the roots of the polynomial function by solving the equation f(x)=0f(x) = 0.
  2. Determine the coefficients of the polynomial function: You can determine the coefficients of the polynomial function by using the relationship between the roots and coefficients.
  3. Determine the nature of the polynomial function: You can determine the nature of the polynomial function by analyzing the coefficients and roots.

Q: What are some common types of polynomial functions?

A: Some common types of polynomial functions include:

  • Linear polynomial functions: These are polynomial functions of degree 1, and have the form f(x)=ax+bf(x) = ax + b.
  • Quadratic polynomial functions: These are polynomial functions of degree 2, and have the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c.
  • Cubic polynomial functions: These are polynomial functions of degree 3, and have the form f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d.

Q: How do I graph a polynomial function?

A: To graph a polynomial function, you can use the following steps:

  1. Determine the degree of the polynomial function: The degree of the polynomial function determines the shape of the graph.
  2. Determine the roots of the polynomial function: The roots of the polynomial function determine the x-intercepts of the graph.
  3. Determine the coefficients of the polynomial function: The coefficients of the polynomial function determine the y-intercept of the graph.
  4. Graph the polynomial function: You can graph the polynomial function by plotting the x-intercepts and y-intercept, and drawing a smooth curve through the points.

Conclusion

In this article, we have answered some frequently asked questions about polynomial functions and their roots. We have discussed the concept of polynomial functions, the relationship between the roots and coefficients, and how to determine the nature of a polynomial function given its roots. We have also discussed how to graph a polynomial function.