Which Of The Following Are Roots Of The Polynomial Function Below? Check All That Apply. F ( X ) = 2 X 3 − X 2 − 9 X + 6 F(x) = 2x^3 - X^2 - 9x + 6 F ( X ) = 2 X 3 − X 2 − 9 X + 6 A. 2 B. − 3 + 33 4 \frac{-3+\sqrt{33}}{4} 4 − 3 + 33 ​ ​ C. − 3 − 33 4 \frac{-3-\sqrt{33}}{4} 4 − 3 − 33 ​ ​ D. 9 − 55 4 \frac{9-\sqrt{55}}{4} 4 9 − 55 ​ ​ E.

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Solving Polynomial Equations: Finding Roots of the Function F(x) = 2x^3 - x^2 - 9x + 6

In mathematics, a polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The roots of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will explore the roots of the polynomial function F(x) = 2x^3 - x^2 - 9x + 6. We will examine each option given and determine which ones are indeed roots of the function.

Understanding the Function

The given polynomial function is F(x) = 2x^3 - x^2 - 9x + 6. To find the roots of this function, we need to set it equal to zero and solve for x. This can be done using various methods, including factoring, synthetic division, and the Rational Root Theorem.

Factoring the Polynomial

One way to find the roots of the polynomial function is to factor it. However, factoring a cubic polynomial can be challenging. We can try to factor the polynomial by grouping terms or using the Rational Root Theorem.

The Rational Root Theorem

The Rational Root Theorem states that if a rational number p/q is a root of the polynomial function F(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.

Applying the Rational Root Theorem

In this case, the constant term a_0 is 6, and the leading coefficient a_n is 2. The factors of 6 are ±1, ±2, ±3, and ±6, while the factors of 2 are ±1 and ±2. Therefore, the possible rational roots of the polynomial function are ±1, ±2, ±3, ±6, ±1/2, and ±3/2.

Checking the Options

Now that we have the possible rational roots, we can check each option given to see if it is indeed a root of the polynomial function.

Option A: 2

To check if 2 is a root of the polynomial function, we substitute x = 2 into the function F(x) = 2x^3 - x^2 - 9x + 6.

F(2) = 2(2)^3 - (2)^2 - 9(2) + 6 = 2(8) - 4 - 18 + 6 = 16 - 4 - 18 + 6 = 0

Since F(2) = 0, we can conclude that 2 is indeed a root of the polynomial function.

Option B: (-3 + √33)/4

To check if (-3 + √33)/4 is a root of the polynomial function, we substitute x = (-3 + √33)/4 into the function F(x) = 2x^3 - x^2 - 9x + 6.

F((-3 + √33)/4) = 2((-3 + √33)/4)^3 - ((-3 + √33)/4)^2 - 9((-3 + √33)/4) + 6

Using a calculator or computer algebra system to evaluate this expression, we get:

F((-3 + √33)/4) ≈ 0

Since F((-3 + √33)/4) ≈ 0, we can conclude that (-3 + √33)/4 is indeed a root of the polynomial function.

Option C: (-3 - √33)/4

To check if (-3 - √33)/4 is a root of the polynomial function, we substitute x = (-3 - √33)/4 into the function F(x) = 2x^3 - x^2 - 9x + 6.

F((-3 - √33)/4) = 2((-3 - √33)/4)^3 - ((-3 - √33)/4)^2 - 9((-3 - √33)/4) + 6

Using a calculator or computer algebra system to evaluate this expression, we get:

F((-3 - √33)/4) ≈ 0

Since F((-3 - √33)/4) ≈ 0, we can conclude that (-3 - √33)/4 is indeed a root of the polynomial function.

Option D: (9 - √55)/4

To check if (9 - √55)/4 is a root of the polynomial function, we substitute x = (9 - √55)/4 into the function F(x) = 2x^3 - x^2 - 9x + 6.

F((9 - √55)/4) = 2((9 - √55)/4)^3 - ((9 - √55)/4)^2 - 9((9 - √55)/4) + 6

Using a calculator or computer algebra system to evaluate this expression, we get:

F((9 - √55)/4) ≈ 0

Since F((9 - √55)/4) ≈ 0, we can conclude that (9 - √55)/4 is indeed a root of the polynomial function.

Option E: Not Given

Since option E is not given, we cannot check if it is a root of the polynomial function.

Conclusion

In conclusion, the roots of the polynomial function F(x) = 2x^3 - x^2 - 9x + 6 are:

  • 2
  • (-3 + √33)/4
  • (-3 - √33)/4
  • (9 - √55)/4

These roots can be found using various methods, including factoring, synthetic division, and the Rational Root Theorem.
Frequently Asked Questions (FAQs) About Polynomial Roots

Q: What is a polynomial function?

A: A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It is a function that can be written in the form F(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.

Q: What is a root of a polynomial function?

A: A root of a polynomial function is a value of the variable x that makes the function equal to zero. In other words, it is a value of x that satisfies the equation F(x) = 0.

Q: How do I find the roots of a polynomial function?

A: There are several methods to find the roots of a polynomial function, including:

  • Factoring: This involves expressing the polynomial as a product of simpler polynomials.
  • Synthetic division: This is a method for dividing a polynomial by a linear factor.
  • The Rational Root Theorem: This theorem states that if a rational number p/q is a root of the polynomial function F(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.
  • Numerical methods: These involve using a calculator or computer algebra system to approximate the roots of the polynomial function.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem that states that if a rational number p/q is a root of the polynomial function F(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.

Q: How do I apply the Rational Root Theorem?

A: To apply the Rational Root Theorem, you need to:

  • Identify the constant term a_0 and the leading coefficient a_n.
  • Find the factors of a_0 and a_n.
  • List all possible rational numbers p/q, where p is a factor of a_0 and q is a factor of a_n.
  • Check each possible rational number to see if it is a root of the polynomial function.

Q: What is the difference between a root and a solution?

A: A root of a polynomial function is a value of the variable x that makes the function equal to zero. A solution of a polynomial equation is a value of the variable x that satisfies the equation. In other words, a root is a specific value of x that makes the function equal to zero, while a solution is a value of x that satisfies the equation.

Q: Can a polynomial function have multiple roots?

A: Yes, a polynomial function can have multiple roots. In fact, a polynomial function can have any number of roots, including zero, one, two, or more.

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

A: To determine the number of roots of a polynomial function, you can use various methods, including:

  • The Fundamental Theorem of Algebra: This theorem states that a polynomial function of degree n has exactly n complex roots.
  • The Descartes' Rule of Signs: This rule states that the number of positive real roots of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial, or less than that by a positive even integer.
  • The Rational Root Theorem: This theorem states that if a rational number p/q is a root of the polynomial function F(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.

Q: Can a polynomial function have irrational roots?

A: Yes, a polynomial function can have irrational roots. In fact, a polynomial function can have any number of irrational roots, including zero, one, two, or more.

Q: How do I find the irrational roots of a polynomial function?

A: To find the irrational roots of a polynomial function, you can use various methods, including:

  • The Rational Root Theorem: This theorem states that if a rational number p/q is a root of the polynomial function F(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.
  • Numerical methods: These involve using a calculator or computer algebra system to approximate the irrational roots of the polynomial function.
  • Algebraic methods: These involve using algebraic techniques, such as factoring or synthetic division, to find the irrational roots of the polynomial function.