Determine Between Which Consecutive Integers The Real Zeros Of F ( X ) = 3 X 3 − 5 X 2 + 5 X + 7 F(x) = 3x^3 - 5x^2 + 5x + 7 F ( X ) = 3 X 3 − 5 X 2 + 5 X + 7 Are Located On The Interval { [-10, 10]$} . I F T H E Z E R O O C C U R S A T A N I N T E G E R , W R I T E T H E I N T E G E R . O P T I O N S : − \[ . If The Zero Occurs At An Integer, Write The Integer.Options:- \[ . I F T H Ezeroocc U Rs A T Anin T E G Er , W R I T E T H E In T E G Er . Opt I O N S : − \[ -1 \ \textless \ X \

Introduction

In this article, we will explore the problem of determining the location of real zeros of a given cubic function, $f(x) = 3x^3 - 5x^2 + 5x + 7$, within a specified interval, $[-10, 10]$. The real zeros of a function are the values of $x$ at which the function intersects the x-axis, resulting in a value of $y = 0$. Our goal is to identify the consecutive integers between which these real zeros lie.

Understanding the Cubic Function

A cubic function is a polynomial function of degree three, which means it has the general form $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants. In our case, the given function is $f(x) = 3x^3 - 5x^2 + 5x + 7$. To determine the real zeros of this function, we need to find the values of $x$ that satisfy the equation $f(x) = 0$.

The Rational Root Theorem

The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$. In our case, the constant term is $7$, and the leading coefficient is $3$. Therefore, the possible rational roots of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$ are the factors of $7$ divided by the factors of $3$.

Finding the Possible Rational Roots

The factors of $7$ are $\pm 1$ and $\pm 7$, and the factors of $3$ are $\pm 1$ and $\pm 3$. Therefore, the possible rational roots of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$ are $\pm 1$, $\pm 7$, $\pm 1/3$, and $\pm 7/3$.

Using Synthetic Division to Test for Rational Roots

To determine whether any of these possible rational roots are actual roots of the function, we can use synthetic division. Synthetic division is a method of dividing a polynomial by a linear factor of the form $(x - r)$, where $r$ is a possible rational root. If the remainder is zero, then $r$ is an actual root of the polynomial.

Testing the Possible Rational Roots

Let's use synthetic division to test each of the possible rational roots.

Testing $x = 1$

	3	-5	5	7
1	3	2	8	14
---	---	---	---	---
	3	2	8

The remainder is not zero, so $x = 1$ is not an actual root of the function.

Testing $x = -1$

	3	-5	5	7
-1	-3	1	-6	-7
---	---	---	---	---
	0	-4	-1	0

The remainder is zero, so $x = -1$ is an actual root of the function.

Testing $x = 7$

	3	-5	5	7
7	21	-14	42	98
---	---	---	---	---
	21	-14	42

The remainder is not zero, so $x = 7$ is not an actual root of the function.

Testing $x = -7$

	3	-5	5	7
-7	-21	35	-105	119
---	---	---	---	---
	-21	35	-105

The remainder is not zero, so $x = -7$ is not an actual root of the function.

Testing $x = 1/3$

	3	-5	5	7
1/3	1	-5/3	5/3	7/3
---	---	---	---	---
	1	-5/3	5/3

The remainder is not zero, so $x = 1/3$ is not an actual root of the function.

Testing $x = -1/3$

	3	-5	5	7
-1/3	-1	5/3	-5/3	-7/3
---	---	---	---	---
	-1	5/3	-5/3

The remainder is not zero, so $x = -1/3$ is not an actual root of the function.

Testing $x = 7/3$

	3	-5	5	7
7/3	7	-35/3	35/3	119/3
---	---	---	---	---
	7	-35/3	35/3

The remainder is not zero, so $x = 7/3$ is not an actual root of the function.

Testing $x = -7/3$

	3	-5	5	7
-7/3	-7	35/3	-35/3	49/3
---	---	---	---	---
	-7	35/3	-35/3

The remainder is not zero, so $x = -7/3$ is not an actual root of the function.

Conclusion

Based on the synthetic division tests, we have found that $x = -1$ is an actual root of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$. To determine the location of the other real zeros, we can use the fact that the function is a cubic function and has at most three real zeros. Since we have already found one real zero, we can use the remaining two zeros to determine the location of the other real zeros.

Finding the Remaining Real Zeros

To find the remaining real zeros, we can use the fact that the function is a cubic function and has at most three real zeros. Since we have already found one real zero, we can use the remaining two zeros to determine the location of the other real zeros.

Let's assume that the remaining two real zeros are $x = a$ and $x = b$. Since the function is a cubic function, we know that the sum of the roots is equal to the negation of the coefficient of the quadratic term divided by the leading coefficient. In this case, the sum of the roots is equal to $5/3$.

We also know that the product of the roots is equal to the constant term divided by the leading coefficient. In this case, the product of the roots is equal to $7/3$.

Using these two facts, we can write the following equations:

$a + b = 5/3$ $ab = 7/3$

Solving these equations, we get:

$a = 2$ $b = 1$

Therefore, the remaining two real zeros are $x = 2$ and $x = 1$.

Conclusion

In conclusion, we have found that the real zeros of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$ are $x = -1$, $x = 2$, and $x = 1$. These zeros lie on the interval $[-10, 10]$, and we have determined the location of the other real zeros using the fact that the function is a cubic function and has at most three real zeros.

Final Answer

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Introduction

In our previous article, we explored the problem of determining the location of real zeros of a given cubic function, $f(x) = 3x^3 - 5x^2 + 5x + 7$, within a specified interval, $[-10, 10]$. We found that the real zeros of the function are $x = -1$, $x = 2$, and $x = 1$. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the significance of the Rational Root Theorem in this problem?

A: The Rational Root Theorem is a fundamental concept in algebra that helps us determine the possible rational roots of a polynomial function. In this problem, we used the Rational Root Theorem to identify the possible rational roots of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$. This theorem is essential in solving polynomial equations and is a crucial tool in algebra.

Q: How do we use synthetic division to test for rational roots?

A: Synthetic division is a method of dividing a polynomial by a linear factor of the form $(x - r)$, where $r$ is a possible rational root. We use synthetic division to test each possible rational root and determine whether it is an actual root of the function. If the remainder is zero, then $r$ is an actual root of the function.

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

A: A rational root is a possible root of a polynomial function, while an actual root is a root that satisfies the equation. In other words, a rational root is a value that could potentially be a root of the function, while an actual root is a value that is actually a root of the function.

Q: How do we determine the location of the other real zeros?

A: To determine the location of the other real zeros, we use the fact that the function is a cubic function and has at most three real zeros. We know that the sum of the roots is equal to the negation of the coefficient of the quadratic term divided by the leading coefficient, and the product of the roots is equal to the constant term divided by the leading coefficient. We can use these two facts to write equations and solve for the remaining real zeros.

Q: What is the importance of understanding the properties of cubic functions?

A: Understanding the properties of cubic functions is essential in solving polynomial equations and is a crucial tool in algebra. Cubic functions have at most three real zeros, and their properties can be used to determine the location of the real zeros. In this problem, we used the properties of cubic functions to determine the location of the real zeros of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$.

Q: How can we apply this knowledge to real-world problems?

A: This knowledge can be applied to real-world problems in various fields, such as physics, engineering, and economics. For example, in physics, we can use cubic functions to model the motion of objects under the influence of gravity. In engineering, we can use cubic functions to design and optimize systems. In economics, we can use cubic functions to model the behavior of economic systems.

Conclusion

In conclusion, determining the location of real zeros of a cubic function is a complex problem that requires a deep understanding of algebra and the properties of cubic functions. We used the Rational Root Theorem, synthetic division, and the properties of cubic functions to determine the location of the real zeros of the function $f(x) = 3x^3 - 5x^2 + 5x + 7$. This knowledge can be applied to real-world problems in various fields and is essential in solving polynomial equations.

Final Answer

The final answer is: $\boxed{-1, 1, 2}$