Finding the Rational Zeros of the Given Function

Oct 13, 2025 by miodeya.com 145 views

Introduction

In mathematics, finding the zeros of a polynomial function is a crucial step in understanding its behavior and properties. A rational zero of a polynomial function is a value of x that makes the function equal to zero, and it is a rational number. In this article, we will explore the concept of rational zeros and use the given function to find all its zeros.

What are Rational Zeros?

A rational zero of a polynomial function is a value of x that makes the function equal to zero, and it is a rational number. Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., p/q, where p and q are integers and q is non-zero. In other words, rational zeros are values of x that make the function equal to zero, and they can be expressed as a fraction.

The Rational Zero Theorem

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

Finding the Rational Zeros of the Given Function

The given function is f(x) = 2x^4 + 2x^3 - 11x^2 + 4x + 3. To find the rational zeros of this function, we need to find the factors of the constant term and the leading coefficient.

The constant term is 3, and its factors are ±1 and ±3.
The leading coefficient is 2, and its factors are ±1 and ±2.

Using the Rational Zero Theorem, we can list all possible combinations of the factors of the constant term and the leading coefficient:

±1/1, ±1/2, ±3/1, ±3/2

These are all possible rational zeros of the given function.

Using Synthetic Division to Find the Zeros

To find the zeros of the given function, we can use synthetic division. Synthetic division is a method of dividing a polynomial function by a linear factor. In this case, we will use synthetic division to divide the given function by each of the possible rational zeros.

Let's start with the first possible rational zero, 1/1 = 1.

2	1	2	-11	4	3
2	2	0	-9	7	3

The result of the synthetic division is 2x^3 + 0x^2 - 9x + 7. This means that 1 is a zero of the given function.

Next, let's try the second possible rational zero, 1/2.

2	1/2	2	-11	4	3
1	1	0	-5	2	3

The result of the synthetic division is x^3 + 0x^2 - 5x + 2. This means that 1/2 is a zero of the given function.

We can continue this process for each of the possible rational zeros. However, we can also use the fact that if a polynomial function has a rational zero, then it must have a corresponding linear factor.

Finding the Linear Factors

Since we know that 1 and 1/2 are zeros of the given function, we can write the corresponding linear factors:

(x - 1) and (x - 1/2)

We can multiply these linear factors to get a quadratic factor:

(x - 1)(2x - 1) = 2x^2 - 3x + 1

This quadratic factor must be a factor of the given function.

Factoring the Given Function

We can use the quadratic factor to factor the given function:

f(x) = 2x^4 + 2x^3 - 11x^2 + 4x + 3

= (2x^2 - 3x + 1)(x^2 + x - 3)

This means that the given function can be factored into two quadratic factors.

Finding the Remaining Zeros

To find the remaining zeros, we can set each of the quadratic factors equal to zero and solve for x.

For the first quadratic factor, 2x^2 - 3x + 1 = 0, we can use the quadratic formula to find the zeros: x = (-b ± √(b^2 - 4ac)) / 2a x = (3 ± √(9 - 8)) / 4 x = (3 ± √1) / 4 x = (3 ± 1) / 4 x = 1 or x = 1/2

We already know that 1 and 1/2 are zeros of the given function.
For the second quadratic factor, x^2 + x - 3 = 0, we can factor it as (x + 3)(x - 1) = 0. This means that x = -3 or x = 1.

We already know that 1 is a zero of the given function.

Conclusion

In this article, we used the Rational Zero Theorem to find the rational zeros of the given function. We then used synthetic division to find the zeros of the function. Finally, we factored the given function into two quadratic factors and found the remaining zeros.

The zeros of the given function are:

x = 1, x = 1/2, x = -3, and x = 3/2

These are the exact values of the zeros, not decimal approximations.

References

"Rational Zero Theorem." Math Open Reference, mathopenref.com/rationalzerotheorem.html.
"Synthetic Division." Math Is Fun, mathisfun.com/algebra/synthetic-division.html.
"Quadratic Formula." Math Is Fun, mathisfun.com/algebra/quadratic-formula.html.
Q&A: The Function Below Has at Least One Rational Zero =====================================================

Q: What is the Rational Zero Theorem?

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

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

A: To find the rational zeros of a polynomial function, you need to find the factors of the constant term and the leading coefficient. Then, you can list all possible combinations of these factors and use synthetic division to find the 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 zeros of a polynomial function.

Q: How do I use synthetic division to find the zeros of a polynomial function?

A: To use synthetic division to find the zeros of a polynomial function, you need to divide the polynomial function by each of the possible rational zeros. The result of the synthetic division will give you a new polynomial function, and you can continue this process until you find all the zeros.

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

A: A rational zero is a value of x that makes the function equal to zero, and it is a rational number. A decimal zero is a value of x that makes the function equal to zero, but it is not a rational number. In other words, a rational zero is a zero that can be expressed as a fraction, while a decimal zero is a zero that cannot be expressed as a fraction.

Q: Can a polynomial function have more than one rational zero?

A: Yes, a polynomial function can have more than one rational zero. In fact, the Rational Zero Theorem states that if a polynomial function has a rational zero, then it must have a corresponding linear factor.

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

A: To find the linear factors of a polynomial function, you need to find the zeros of the function. Then, you can write the corresponding linear factors and multiply them to get a quadratic factor.

Q: Can a polynomial function have a quadratic factor that is not a factor of the original function?

A: No, a polynomial function cannot have a quadratic factor that is not a factor of the original function. If a quadratic factor is a factor of the original function, then it must be a factor of the function.

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

A: To find the remaining zeros of a polynomial function, you need to set each of the quadratic factors equal to zero and solve for x. This will give you the remaining zeros of the function.

Q: Can a polynomial function have a zero that is not a rational number?

A: Yes, a polynomial function can have a zero that is not a rational number. In fact, the zeros of a polynomial function can be irrational numbers, complex numbers, or even transcendental numbers.

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

A: To know if a polynomial function has a rational zero, you need to use the Rational Zero Theorem. This theorem states that if a polynomial function has a rational zero, then it must have a corresponding linear factor.

Q: Can a polynomial function have a rational zero that is not a factor of the original function?

A: No, a polynomial function cannot have a rational zero that is not a factor of the original function. If a rational zero is a factor of the original function, then it must be a factor of the function.

Q: How do I find the exact values of the zeros of a polynomial function?

A: To find the exact values of the zeros of a polynomial function, you need to use the quadratic formula or synthetic division. These methods will give you the exact values of the zeros, not decimal approximations.

Q: Can a polynomial function have a zero that is a complex number?

A: Yes, a polynomial function can have a zero that is a complex number. In fact, the zeros of a polynomial function can be complex numbers, irrational numbers, or even transcendental numbers.

Q: How do I know if a polynomial function has a complex zero?

A: To know if a polynomial function has a complex zero, you need to use the quadratic formula or synthetic division. These methods will give you the complex zeros of the function.

Q: Can a polynomial function have a zero that is a transcendental number?

A: Yes, a polynomial function can have a zero that is a transcendental number. In fact, the zeros of a polynomial function can be complex numbers, irrational numbers, or even transcendental numbers.

Q: How do I know if a polynomial function has a transcendental zero?

A: To know if a polynomial function has a transcendental zero, you need to use the quadratic formula or synthetic division. These methods will give you the transcendental zeros of the function.

Conclusion

In this Q&A article, we have discussed the Rational Zero Theorem, synthetic division, and the different types of zeros that a polynomial function can have. We have also answered some common questions about polynomial functions and their zeros.