What Are All The Rational Roots Of The Polynomial $f(x) = 20x^4 + X^3 + 8x^2 + X - 12$?A. $-\frac{4}{5}$ And $\frac{3}{4}$B. $-\frac{4}{5}$ And $-\frac{3}{4}$C. $-1, -\frac{4}{5}, \frac{3}{4},$ And

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Introduction

In mathematics, polynomials are a fundamental concept in algebra, and understanding their roots is crucial for solving various mathematical problems. The rational roots theorem is a powerful tool for finding the rational roots of a polynomial. In this article, we will explore the rational roots of the polynomial f(x)=20x4+x3+8x2+x−12f(x) = 20x^4 + x^3 + 8x^2 + x - 12.

What are Rational Roots?

Rational roots are the roots of a polynomial that can be expressed as a fraction of two integers, i.e., pq\frac{p}{q}, where pp and qq are integers and qq is non-zero. Rational roots can be positive or negative and can be expressed in the form of ab\frac{a}{b}, where aa and bb are integers.

The Rational Roots Theorem

The rational roots theorem states that if a rational number p/qp/q is a root of the polynomial f(x)f(x), then pp must be a factor of the constant term of the polynomial, and qq must be a factor of the leading coefficient of the polynomial. In other words, if p/qp/q is a root of the polynomial, then pp divides the constant term, and qq divides the leading coefficient.

Finding the Rational Roots of the Polynomial

To find the rational roots of the polynomial f(x)=20x4+x3+8x2+x−12f(x) = 20x^4 + x^3 + 8x^2 + x - 12, we need to find the factors of the constant term and the leading coefficient. The constant term is −12-12, and its factors are ±1,±2,±3,±4,±6,±12\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12. The leading coefficient is 2020, and its factors are ±1,±2,±4,±5,±10,±20\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20.

Using the Rational Roots Theorem

Using the rational roots theorem, we can list all possible rational roots of the polynomial. The possible rational roots are:

  • ±11\pm \frac{1}{1}
  • ±12\pm \frac{1}{2}
  • ±14\pm \frac{1}{4}
  • ±15\pm \frac{1}{5}
  • ±110\pm \frac{1}{10}
  • ±120\pm \frac{1}{20}
  • ±21\pm \frac{2}{1}
  • ±22\pm \frac{2}{2}
  • ±24\pm \frac{2}{4}
  • ±25\pm \frac{2}{5}
  • ±210\pm \frac{2}{10}
  • ±220\pm \frac{2}{20}
  • ±31\pm \frac{3}{1}
  • ±32\pm \frac{3}{2}
  • ±34\pm \frac{3}{4}
  • ±35\pm \frac{3}{5}
  • ±310\pm \frac{3}{10}
  • ±320\pm \frac{3}{20}
  • ±41\pm \frac{4}{1}
  • ±42\pm \frac{4}{2}
  • ±44\pm \frac{4}{4}
  • ±45\pm \frac{4}{5}
  • ±410\pm \frac{4}{10}
  • ±420\pm \frac{4}{20}
  • ±61\pm \frac{6}{1}
  • ±62\pm \frac{6}{2}
  • ±64\pm \frac{6}{4}
  • ±65\pm \frac{6}{5}
  • ±610\pm \frac{6}{10}
  • ±620\pm \frac{6}{20}
  • ±121\pm \frac{12}{1}
  • ±122\pm \frac{12}{2}
  • ±124\pm \frac{12}{4}
  • ±125\pm \frac{12}{5}
  • ±1210\pm \frac{12}{10}
  • ±1220\pm \frac{12}{20}

Simplifying the List of Possible Rational Roots

We can simplify the list of possible rational roots by canceling out any common factors between the numerator and the denominator. For example, 22\frac{2}{2} can be simplified to 11. Similarly, 44\frac{4}{4} can be simplified to 11.

Finding the Rational Roots of the Polynomial

To find the rational roots of the polynomial, we need to test each possible rational root by substituting it into the polynomial and checking if the result is equal to zero. We can use synthetic division or long division to divide the polynomial by each possible rational root.

Testing the Possible Rational Roots

Let's test each possible rational root by substituting it into the polynomial and checking if the result is equal to zero.

  • 11\frac{1}{1}: f(1)=20(1)4+(1)3+8(1)2+(1)−12=20+1+8+1−12=18≠0f(1) = 20(1)^4 + (1)^3 + 8(1)^2 + (1) - 12 = 20 + 1 + 8 + 1 - 12 = 18 \neq 0
  • 12\frac{1}{2}: f(12)=20(12)4+(12)3+8(12)2+(12)−12=20(116)+18+8(14)+12−12=54+18+2+12−12=−478≠0f(\frac{1}{2}) = 20(\frac{1}{2})^4 + (\frac{1}{2})^3 + 8(\frac{1}{2})^2 + (\frac{1}{2}) - 12 = 20(\frac{1}{16}) + \frac{1}{8} + 8(\frac{1}{4}) + \frac{1}{2} - 12 = \frac{5}{4} + \frac{1}{8} + 2 + \frac{1}{2} - 12 = -\frac{47}{8} \neq 0
  • 14\frac{1}{4}: f(14)=20(14)4+(14)3+8(14)2+(14)−12=20(1256)+164+8(116)+14−12=5128+164+12+14−12=−947128≠0f(\frac{1}{4}) = 20(\frac{1}{4})^4 + (\frac{1}{4})^3 + 8(\frac{1}{4})^2 + (\frac{1}{4}) - 12 = 20(\frac{1}{256}) + \frac{1}{64} + 8(\frac{1}{16}) + \frac{1}{4} - 12 = \frac{5}{128} + \frac{1}{64} + \frac{1}{2} + \frac{1}{4} - 12 = -\frac{947}{128} \neq 0
  • 15\frac{1}{5}: f(15)=20(15)4+(15)3+8(15)2+(15)−12=20(1625)+1125+8(125)+15−12=5625+1125+825+15−12=−946625≠0f(\frac{1}{5}) = 20(\frac{1}{5})^4 + (\frac{1}{5})^3 + 8(\frac{1}{5})^2 + (\frac{1}{5}) - 12 = 20(\frac{1}{625}) + \frac{1}{125} + 8(\frac{1}{25}) + \frac{1}{5} - 12 = \frac{5}{625} + \frac{1}{125} + \frac{8}{25} + \frac{1}{5} - 12 = -\frac{946}{625} \neq 0
  • 110\frac{1}{10}: f(110)=20(110)4+(110)3+8(110)2+(110)−12=20(110000)+11000+8(1100)+110−12=510000+11000+8100+110−12=−946110000≠0f(\frac{1}{10}) = 20(\frac{1}{10})^4 + (\frac{1}{10})^3 + 8(\frac{1}{10})^2 + (\frac{1}{10}) - 12 = 20(\frac{1}{10000}) + \frac{1}{1000} + 8(\frac{1}{100}) + \frac{1}{10} - 12 = \frac{5}{10000} + \frac{1}{1000} + \frac{8}{100} + \frac{1}{10} - 12 = -\frac{9461}{10000} \neq 0
  • 120\frac{1}{20}: f(120)=20(120)4+(120)3+8(120)2+(120)−12=20(1160000)+18000+8(1400)+120−12=5160000+18000+8400+120−12=−94611160000≠0f(\frac{1}{20}) = 20(\frac{1}{20})^4 + (\frac{1}{20})^3 + 8(\frac{1}{20})^2 + (\frac{1}{20}) - 12 = 20(\frac{1}{160000}) + \frac{1}{8000} + 8(\frac{1}{400}) + \frac{1}{20} - 12 = \frac{5}{160000} + \frac{1}{8000} + \frac{8}{400} + \frac{1}{20} - 12 = -\frac{94611}{160000} \neq 0
  • 21\frac{2}{1}: f(2)=20(2)4+(2)3+8(2)2+(2)−12=20(16)+8+32+2−12=320+8+32+2−12=350≠0f(2) = 20(2)^4 + (2)^3 + 8(2)^2 + (2) - 12 = 20(16) + 8 + 32 + 2 - 12 = 320 + 8 + 32 + 2 - 12 = 350 \neq 0
  • 22\frac{2}{2}: f(1)=20(1)4+(1)3+8(1)2+(1)−12=20+1+8+1−12=18≠0f(1) = 20(1)^4 + (1)^3 + 8(1)^2 + (1) - 12 = 20 + 1 + 8 + 1 - 12 = 18 \neq 0
  • 24\frac{2}{4}: $f(\frac{1}{2}) = 20(\frac{1}{2})

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Q&A: Rational Roots of a Polynomial

Q: What is the rational roots theorem?

A: The rational roots theorem is a powerful tool for finding the rational roots of a polynomial. It states that if a rational number p/qp/q is a root of the polynomial f(x)f(x), then pp must be a factor of the constant term of the polynomial, and qq must be a factor of the leading coefficient of the polynomial.

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

A: To find the rational roots of a polynomial, you need to find the factors of the constant term and the leading coefficient. Then, you can use the rational roots theorem to list all possible rational roots. Finally, you can test each possible rational root by substituting it into the polynomial and checking if the result is equal to zero.

Q: What are the possible rational roots of the polynomial f(x)=20x4+x3+8x2+x−12f(x) = 20x^4 + x^3 + 8x^2 + x - 12?

A: The possible rational roots of the polynomial f(x)=20x4+x3+8x2+x−12f(x) = 20x^4 + x^3 + 8x^2 + x - 12 are:

  • ±11\pm \frac{1}{1}
  • ±12\pm \frac{1}{2}
  • ±14\pm \frac{1}{4}
  • ±15\pm \frac{1}{5}
  • ±110\pm \frac{1}{10}
  • ±120\pm \frac{1}{20}
  • ±21\pm \frac{2}{1}
  • ±22\pm \frac{2}{2}
  • ±24\pm \frac{2}{4}
  • ±25\pm \frac{2}{5}
  • ±210\pm \frac{2}{10}
  • ±220\pm \frac{2}{20}
  • ±31\pm \frac{3}{1}
  • ±32\pm \frac{3}{2}
  • ±34\pm \frac{3}{4}
  • ±35\pm \frac{3}{5}
  • ±310\pm \frac{3}{10}
  • ±320\pm \frac{3}{20}
  • ±41\pm \frac{4}{1}
  • ±42\pm \frac{4}{2}
  • ±44\pm \frac{4}{4}
  • ±45\pm \frac{4}{5}
  • ±410\pm \frac{4}{10}
  • ±420\pm \frac{4}{20}
  • ±61\pm \frac{6}{1}
  • ±62\pm \frac{6}{2}
  • ±64\pm \frac{6}{4}
  • ±65\pm \frac{6}{5}
  • ±610\pm \frac{6}{10}
  • ±620\pm \frac{6}{20}
  • ±121\pm \frac{12}{1}
  • ±122\pm \frac{12}{2}
  • ±124\pm \frac{12}{4}
  • ±125\pm \frac{12}{5}
  • ±1210\pm \frac{12}{10}
  • ±1220\pm \frac{12}{20}

Q: How do I test each possible rational root?

A: To test each possible rational root, you need to substitute it into the polynomial and check if the result is equal to zero. You can use synthetic division or long division to divide the polynomial by each possible rational root.

Q: What are the rational roots of the polynomial f(x)=20x4+x3+8x2+x−12f(x) = 20x^4 + x^3 + 8x^2 + x - 12?

A: The rational roots of the polynomial f(x)=20x4+x3+8x2+x−12f(x) = 20x^4 + x^3 + 8x^2 + x - 12 are:

  • −1-1
  • −45-\frac{4}{5}
  • 34\frac{3}{4}

Q: How do I use the rational roots theorem to find the rational roots of a polynomial?

A: To use the rational roots theorem to find the rational roots of a polynomial, you need to follow these steps:

  1. Find the factors of the constant term and the leading coefficient.
  2. List all possible rational roots using the rational roots theorem.
  3. Test each possible rational root by substituting it into the polynomial and checking if the result is equal to zero.
  4. The rational roots of the polynomial are the values that make the polynomial equal to zero.

Q: What are the applications of the rational roots theorem?

A: The rational roots theorem has many applications in mathematics and science. It is used to find the rational roots of polynomials, which is essential in solving various mathematical problems. It is also used in science to model real-world phenomena and make predictions.

Q: How do I use the rational roots theorem to solve real-world problems?

A: To use the rational roots theorem to solve real-world problems, you need to follow these steps:

  1. Identify the problem and the polynomial that models it.
  2. Find the factors of the constant term and the leading coefficient.
  3. List all possible rational roots using the rational roots theorem.
  4. Test each possible rational root by substituting it into the polynomial and checking if the result is equal to zero.
  5. The rational roots of the polynomial are the values that make the polynomial equal to zero, which can be used to solve the problem.

Q: What are the limitations of the rational roots theorem?

A: The rational roots theorem has some limitations. It only works for polynomials with rational coefficients, and it may not work for polynomials with complex or irrational coefficients. Additionally, it may not work for polynomials with multiple roots or repeated roots.