Finding Possible Rational Zeros

Oct 13, 2025 by wbrum.org 57 views

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Introduction

In this article, we will explore the process of finding rational zeros of a polynomial function. Rational zeros are the values of x that make the polynomial function equal to zero. We will use the Rational Zero Theorem to find the possible rational zeros of the given polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ . Then, we will use the depressed equation to find all roots of the equation $f(x)=0$ .

The Rational Zero Theorem

The Rational Zero Theorem states that if a rational number p/q is a zero of the polynomial function f(x), then p must be a factor of the constant term, and q must be a factor of the leading coefficient. In other words, the possible rational zeros of a polynomial function are all the possible combinations of the factors of the constant term and the factors of the leading coefficient.

Finding Possible Rational Zeros

To find the possible rational zeros of the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ , we need to find the factors of the constant term -2 and the factors of the leading coefficient 4.

Factors of the Constant Term

The factors of the constant term -2 are: ±1, ±2.

Factors of the Leading Coefficient

The factors of the leading coefficient 4 are: ±1, ±2, ±4.

Possible Rational Zeros

Using the Rational Zero Theorem, we can find the possible rational zeros of the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ by combining the factors of the constant term and the factors of the leading coefficient.

±1/1 = ±1
±1/2 = ±0.5
±2/1 = ±2
±2/2 = ±1
±2/4 = ±0.5
±4/1 = ±4
±4/2 = ±2
±4/4 = ±1

Reducing the List of Possible Rational Zeros

We can reduce the list of possible rational zeros by eliminating any duplicates.

±1
±0.5
±2
±4

Using Synthetic Division to Find Rational Zeros

To find the rational zeros of the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ , we can use synthetic division to test each possible rational zero.

Synthetic Division

Synthetic division is a method of dividing a polynomial function by a linear factor. It is a shortcut for long division and can be used to find the quotient and remainder of a polynomial function.

Testing Possible Rational Zeros

We can use synthetic division to test each possible rational zero.

x = 1: Using synthetic division, we find that x = 1 is not a rational zero.
x = -1: Using synthetic division, we find that x = -1 is a rational zero.
x = 0.5: Using synthetic division, we find that x = 0.5 is not a rational zero.
x = -0.5: Using synthetic division, we find that x = -0.5 is not a rational zero.
x = 2: Using synthetic division, we find that x = 2 is not a rational zero.
x = -2: Using synthetic division, we find that x = -2 is not a rational zero.
x = 4: Using synthetic division, we find that x = 4 is not a rational zero.
x = -4: Using synthetic division, we find that x = -4 is not a rational zero.

Finding the Depressed Equation

The depressed equation is a quadratic equation that is obtained by dividing the polynomial function by the linear factor that corresponds to the rational zero.

Dividing the Polynomial Function

We can divide the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ by the linear factor x + 1 to obtain the depressed equation.

Quotient: 4x^2 - 8x - 2
Remainder: 0

Solving the Depressed Equation

The depressed equation is a quadratic equation that can be solved using the quadratic formula.

Quadratic Formula

The quadratic formula is a method of solving quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Solving the Depressed Equation

We can use the quadratic formula to solve the depressed equation.

a = 4
b = -8
c = -2

x = (8 ± √((-8)^2 - 4(4)(-2))) / 2(4) x = (8 ± √(64 + 32)) / 8 x = (8 ± √96) / 8 x = (8 ± 4√6) / 8 x = 1 ± √6/2

Conclusion

In this article, we have found the rational zeros of the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ using the Rational Zero Theorem and synthetic division. We have also used the depressed equation to find all roots of the equation $f(x)=0$ . The rational zeros of the polynomial function are x = -1 and x = 1 ± √6/2.

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Q: What is the Rational Zero Theorem?

A: The Rational Zero Theorem is a theorem that states that if a rational number p/q is a zero of the polynomial function f(x), then p must be a factor of the constant term, and q must be a factor of the leading coefficient.

Q: How do I find the possible rational zeros of a polynomial function?

A: To find the possible rational zeros of a polynomial function, you need to find the factors of the constant term and the factors of the leading coefficient. Then, you can combine these factors to find the possible rational zeros.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial function by a linear factor. It is a shortcut for long division and can be used to find the quotient and remainder of a polynomial function.

Q: How do I use synthetic division to find rational zeros?

A: To use synthetic division to find rational zeros, you need to test each possible rational zero by dividing the polynomial function by the linear factor that corresponds to the rational zero.

Q: What is the depressed equation?

A: The depressed equation is a quadratic equation that is obtained by dividing the polynomial function by the linear factor that corresponds to the rational zero.

Q: How do I solve the depressed equation?

A: To solve the depressed equation, you can use the quadratic formula. The quadratic formula is a method of solving quadratic equations and is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: What are the rational zeros of the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ ?

A: The rational zeros of the polynomial function $f(x) = 4x^3 - 12x^2 - 11x - 2$ are x = -1 and x = 1 ± √6/2.

Q: How do I find the roots of the equation $f(x)=0$ ?

A: To find the roots of the equation $f(x)=0$ , you need to find the rational zeros of the polynomial function and then use the depressed equation to find all roots of the equation.

Q: What is the importance of finding rational zeros of a polynomial function?

A: Finding rational zeros of a polynomial function is important because it helps us to factor the polynomial function and find all roots of the equation. This can be useful in a variety of applications, such as solving systems of equations and finding the maximum or minimum value of a function.

Q: Can I use the Rational Zero Theorem to find irrational zeros of a polynomial function?

A: No, the Rational Zero Theorem only applies to rational zeros of a polynomial function. To find irrational zeros, you need to use other methods, such as the quadratic formula or numerical methods.

Q: How do I know if a polynomial function has any rational zeros?

A: You can use the Rational Zero Theorem to determine if a polynomial function has any rational zeros. If the constant term and the leading coefficient have any common factors, then the polynomial function may have rational zeros.

Q: Can I use synthetic division to find irrational zeros of a polynomial function?

A: No, synthetic division is only used to find rational zeros of a polynomial function. To find irrational zeros, you need to use other methods, such as the quadratic formula or numerical methods.

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

A: To find the depressed equation of a polynomial function, you need to divide the polynomial function by the linear factor that corresponds to the rational zero.

Q: Can I use the depressed equation to find all roots of the equation $f(x)=0$ ?

A: Yes, the depressed equation can be used to find all roots of the equation $f(x)=0$ . However, you need to use the quadratic formula to solve the depressed equation.

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

A: A rational zero is a zero of a polynomial function that is a rational number, i.e., a number that can be expressed as the ratio of two integers. An irrational zero is a zero of a polynomial function that is an irrational number, i.e., a number that cannot be expressed as the ratio of two integers.