The Function Below Has At Least One Rational Root. Find The $y$-intercept And Use The Rational Roots Theorem To Find All Rational Roots. Fill In The Sign Table And Sketch A Graph Below. Your Graph Must Accurately Cross All Known

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Introduction

In mathematics, the rational roots theorem is a fundamental concept used to find the rational roots of a polynomial equation. This theorem states that if a polynomial equation has a rational root, then that root must be a factor of the constant term divided by a factor of the leading coefficient. In this article, we will explore the function below, find the y-intercept, and use the rational roots theorem to find all rational roots. We will also fill in the sign table and sketch a graph that accurately crosses all known roots.

The Function

Let's consider the function:

f(x) = x^3 + 2x^2 - 7x - 12

This is a cubic polynomial function, and we are asked to find the y-intercept and use the rational roots theorem to find all rational roots.

Finding the y-Intercept

The y-intercept is the point where the graph of the function crosses the y-axis. To find the y-intercept, we need to evaluate the function at x = 0.

f(0) = (0)^3 + 2(0)^2 - 7(0) - 12 f(0) = -12

Therefore, the y-intercept of the function is -12.

Using the Rational Roots Theorem

The rational roots theorem states that if a polynomial equation has a rational root, then that root must be a factor of the constant term divided by a factor of the leading coefficient. In this case, the constant term is -12, and the leading coefficient is 1.

The factors of -12 are: ±1, ±2, ±3, ±4, ±6, and ±12.

The factors of 1 are: ±1.

Therefore, the possible rational roots of the function are: ±1, ±2, ±3, ±4, ±6, and ±12.

Filling in the Sign Table

To find the rational roots of the function, we need to fill in the sign table. The sign table is a table that shows the sign of the function at different points.

x f(x)
-∞ -
-3 ?
-2 ?
-1 ?
0 -12
1 ?
2 ?
3 ?
∞ +

To fill in the sign table, we need to evaluate the function at different points.

f(-3) = (-3)^3 + 2(-3)^2 - 7(-3) - 12 f(-3) = -27 + 18 + 21 - 12 f(-3) = 0

Therefore, the function crosses the x-axis at x = -3.

f(-2) = (-2)^3 + 2(-2)^2 - 7(-2) - 12 f(-2) = -8 + 8 + 14 - 12 f(-2) = 2

Therefore, the function does not cross the x-axis at x = -2.

f(-1) = (-1)^3 + 2(-1)^2 - 7(-1) - 12 f(-1) = -1 + 2 + 7 - 12 f(-1) = -4

Therefore, the function does not cross the x-axis at x = -1.

f(1) = (1)^3 + 2(1)^2 - 7(1) - 12 f(1) = 1 + 2 - 7 - 12 f(1) = -16

Therefore, the function does not cross the x-axis at x = 1.

f(2) = (2)^3 + 2(2)^2 - 7(2) - 12 f(2) = 8 + 8 - 14 - 12 f(2) = -10

Therefore, the function does not cross the x-axis at x = 2.

f(3) = (3)^3 + 2(3)^2 - 7(3) - 12 f(3) = 27 + 18 - 21 - 12 f(3) = 12

Therefore, the function does not cross the x-axis at x = 3.

Sketching the Graph

Based on the sign table, we can sketch the graph of the function.

The graph of the function crosses the x-axis at x = -3 and x = 0.

The graph of the function does not cross the x-axis at x = -2, -1, 1, 2, and 3.

The graph of the function is a cubic polynomial function, and it has a y-intercept at -12.

Conclusion

In this article, we have found the y-intercept of the function and used the rational roots theorem to find all rational roots. We have also filled in the sign table and sketched a graph that accurately crosses all known roots.

The rational roots theorem is a powerful tool used to find the rational roots of a polynomial equation. By using this theorem, we can find the rational roots of a function and sketch a graph that accurately crosses all known roots.

References

Further Reading

Introduction

In our previous article, we explored the function f(x) = x^3 + 2x^2 - 7x - 12 and found the y-intercept and used the rational roots theorem to find all rational roots. We also filled in the sign table and sketched a graph that accurately crosses all known roots. In this article, we will answer some frequently asked questions related to the rational roots theorem and the function f(x) = x^3 + 2x^2 - 7x - 12.

Q&A

Q: What is the rational roots theorem?

A: The rational roots theorem is a fundamental concept in mathematics that states that if a polynomial equation has a rational root, then that root must be a factor of the constant term divided by a factor of the leading coefficient.

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

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

  1. Find the factors of the constant term.
  2. Find the factors of the leading coefficient.
  3. Divide the factors of the constant term by the factors of the leading coefficient.
  4. The resulting fractions are the possible rational roots of the polynomial equation.

Q: What is the difference between a rational root and an irrational root?

A: A rational root is a root that can be expressed as a fraction of two integers, while an irrational root is a root that cannot be expressed as a fraction of two integers.

Q: Can a polynomial equation have more than one rational root?

A: Yes, a polynomial equation can have more than one rational root. For example, the polynomial equation x^2 - 4 = 0 has two rational roots: x = 2 and x = -2.

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

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

  1. Find the factors of the constant term.
  2. Find the factors of the leading coefficient.
  3. Divide the factors of the constant term by the factors of the leading coefficient.
  4. The resulting fractions are the possible rational roots of the polynomial equation.
  5. Test each possible rational root by substituting it into the polynomial equation.
  6. If the polynomial equation equals zero when the possible rational root is substituted, then that root is a rational root of the polynomial equation.

Q: What is the significance of the rational roots theorem?

A: The rational roots theorem is a powerful tool used to find the rational roots of a polynomial equation. By using this theorem, you can find the rational roots of a function and sketch a graph that accurately crosses all known roots.

Q: Can the rational roots theorem be used to find the irrational roots of a polynomial equation?

A: No, the rational roots theorem can only be used to find the rational roots of a polynomial equation. To find the irrational roots of a polynomial equation, you need to use other methods such as the quadratic formula or numerical methods.

Q: How do I sketch a graph that accurately crosses all known roots?

A: To sketch a graph that accurately crosses all known roots, you need to follow these steps:

  1. Find the rational roots of the polynomial equation using the rational roots theorem.
  2. Find the y-intercept of the polynomial equation.
  3. Plot the rational roots and the y-intercept on a graph.
  4. Draw a smooth curve that passes through the rational roots and the y-intercept.

Conclusion

In this article, we have answered some frequently asked questions related to the rational roots theorem and the function f(x) = x^3 + 2x^2 - 7x - 12. We hope that this article has provided you with a better understanding of the rational roots theorem and how to use it to find the rational roots of a polynomial equation.

References

Further Reading