Find A Polynomial Of Degree $n$ That Has The Given Zero(s).Zero(s): $x = 0, -\sqrt{7}, \sqrt{7}$Degree: $n = 5$$f(x) = \square$

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Understanding the Problem

When given a set of zeros, we can find a polynomial of degree $n$ that has those zeros. In this case, we are given three zeros: $x = 0, -\sqrt{7}, \sqrt{7}$, and we need to find a polynomial of degree $n = 5$ that has these zeros.

The Factor Theorem

The factor theorem states that if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$. This means that if we have a zero $a$, we can write a factor of the polynomial as $(x - a)$.

Writing the Factors

Using the factor theorem, we can write the factors of the polynomial as:

• $(x - 0)$ or simply $x$
• $(x + \sqrt{7})$
• $(x - \sqrt{7})$

Multiplying the Factors

To find the polynomial, we need to multiply these factors together. Since we have three factors, we will have a polynomial of degree $3$. However, we are given that the degree of the polynomial is $5$, so we need to multiply the factors by a quadratic expression to get a polynomial of degree $5$.

Finding the Quadratic Expression

Let's assume the quadratic expression is $ax^2 + bx + c$. We need to find the values of $a$, $b$, and $c$ such that the product of the factors and the quadratic expression is a polynomial of degree $5$.

The Product of the Factors

The product of the factors is:

$$x(x + \sqrt{7})(x - \sqrt{7})$$

Expanding the Product

Expanding the product, we get:

$$x(x^2 - 7)$$

Multiplying by the Quadratic Expression

Multiplying the product by the quadratic expression, we get:

$$x(x^2 - 7)(ax^2 + bx + c)$$

Expanding the Product

Expanding the product, we get:

$$ax^5 + bx^4 + cx^3 - 7ax^3 - 7bx^2 - 7cx$$

Combining Like Terms

Combining like terms, we get:

$$ax^5 + (b - 7a)x^4 + (c - 7b)x^3 - 7cx$$

Finding the Values of $a$, $b$, and $c$

We need to find the values of $a$, $b$, and $c$ such that the polynomial has the given zeros. Since the polynomial has a zero at $x = 0$, we know that $a \neq 0$. We also know that the polynomial has a zero at $x = -\sqrt{7}$ and $x = \sqrt{7}$, so we can write:

$$(-\sqrt{7})^5 + (b - 7a)(-\sqrt{7})^4 + (c - 7b)(-\sqrt{7})^3 - 7c(-\sqrt{7}) = 0$$

$$\sqrt{7}^5 + (b - 7a)\sqrt{7}^4 + (c - 7b)\sqrt{7}^3 - 7c\sqrt{7} = 0$$

Solving for $a$, $b$, and $c$

Solving the system of equations, we get:

$$a = 1$$

$$b = 0$$

$$c = 0$$

The Polynomial

Substituting the values of $a$, $b$, and $c$ into the polynomial, we get:

$$f(x) = x(x^2 - 7)$$

Simplifying the Polynomial

Simplifying the polynomial, we get:

$$f(x) = x^3 - 7x$$

Conclusion

In this article, we found a polynomial of degree $n = 5$ that has the given zeros: $x = 0, -\sqrt{7}, \sqrt{7}$. The polynomial is:

$$f(x) = x^3 - 7x$$

This polynomial satisfies the given conditions and has the desired zeros.

Example Use Case

Suppose we want to find the polynomial that has the zeros $x = 0, -2, 2$. We can follow the same steps as above to find the polynomial.

Step 1: Write the Factors

The factors of the polynomial are:

• $(x - 0)$ or simply $x$
• $(x + 2)$
• $(x - 2)$

Step 2: Multiply the Factors

Multiplying the factors together, we get:

$$x(x + 2)(x - 2)$$

Step 3: Expand the Product

Expanding the product, we get:

$$x(x^2 - 4)$$

Step 4: Multiply by the Quadratic Expression

Multiplying the product by the quadratic expression, we get:

$$x(x^2 - 4)(ax^2 + bx + c)$$

Step 5: Expand the Product

Expanding the product, we get:

$$ax^5 + bx^4 + cx^3 - 4ax^3 - 4bx^2 - 4cx$$

Step 6: Combine Like Terms

Combining like terms, we get:

$$ax^5 + (b - 4a)x^4 + (c - 4b)x^3 - 4cx$$

Step 7: Find the Values of $a$, $b$, and $c$

We need to find the values of $a$, $b$, and $c$ such that the polynomial has the given zeros. Since the polynomial has a zero at $x = 0$, we know that $a \neq 0$. We also know that the polynomial has a zero at $x = -2$ and $x = 2$, so we can write:

$$(-2)^5 + (b - 4a)(-2)^4 + (c - 4b)(-2)^3 - 4c(-2) = 0$$

$$2^5 + (b - 4a)2^4 + (c - 4b)2^3 - 4c2 = 0$$

Step 8: Solve for $a$, $b$, and $c$

Solving the system of equations, we get:

$$a = 1$$

$$b = 0$$

$$c = 0$$

Step 9: The Polynomial

Substituting the values of $a$, $b$, and $c$ into the polynomial, we get:

$$f(x) = x(x^2 - 4)$$

Step 10: Simplify the Polynomial

Simplifying the polynomial, we get:

$$f(x) = x^3 - 4x$$

This polynomial satisfies the given conditions and has the desired zeros.

Conclusion

In this article, we found a polynomial of degree $n = 5$ that has the given zeros: $x = 0, -2, 2$. The polynomial is:

$$f(x) = x^3 - 4x$$

This polynomial satisfies the given conditions and has the desired zeros.

References

• [1] "The Factor Theorem" by Math Open Reference
• [2] "Polynomial Factorization" by Wolfram MathWorld
• [3] "Polynomial Roots" by Math Is Fun

Glossary

• Factor Theorem: If $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$.
• Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Degree: The highest power of the variable in a polynomial.
• Zero: A value of the variable that makes the polynomial equal to zero.
  

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Q: What is the factor theorem?

A: The factor theorem states that if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$. This means that if we have a zero $a$, we can write a factor of the polynomial as $(x - a)$.

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

A: To find the factors of a polynomial, we need to identify the zeros of the polynomial. If we have a zero $a$, we can write a factor of the polynomial as $(x - a)$. We can then multiply these factors together to get the polynomial.

Q: What is the difference between a zero and a root?

A: A zero and a root are the same thing. A zero is a value of the variable that makes the polynomial equal to zero.

Q: How do I find the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $x^3 + 2x^2 + 3x + 4$, the degree is 3.

Q: What is the relationship between the degree of a polynomial and the number of zeros?

A: The degree of a polynomial is equal to the number of zeros, minus 1. For example, if a polynomial has 5 zeros, its degree is 4.

Q: How do I find the polynomial with given zeros?

A: To find the polynomial with given zeros, we need to multiply the factors of the polynomial together. If we have $n$ zeros, we will have a polynomial of degree $n-1$.

Q: What is the quadratic expression?

A: The quadratic expression is a polynomial of degree 2. It is used to multiply the factors of the polynomial together to get a polynomial of degree $n$.

Q: How do I find the values of $a$, $b$, and $c$ in the quadratic expression?

A: To find the values of $a$, $b$, and $c$ in the quadratic expression, we need to solve a system of equations. We can use the given zeros to write equations and then solve for $a$, $b$, and $c$.

Q: What is the final polynomial?

A: The final polynomial is the product of the factors and the quadratic expression. It is a polynomial of degree $n$ that has the given zeros.

Q: How do I simplify the polynomial?

A: To simplify the polynomial, we need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

Q: What is the importance of finding a polynomial with given zeros?

A: Finding a polynomial with given zeros is important in many areas of mathematics and science. It is used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: Can I use a calculator to find the polynomial?

A: Yes, you can use a calculator to find the polynomial. However, it is often more efficient and effective to use a computer algebra system (CAS) or a graphing calculator.

Q: What are some common applications of polynomials?

A: Polynomials have many common applications in mathematics and science. Some examples include:

• Modeling the motion of objects
• Describing the growth of populations
• Analyzing the behavior of electrical circuits
• Solving systems of equations
• Finding the roots of a polynomial

Q: Can I use polynomials to solve real-world problems?

A: Yes, you can use polynomials to solve real-world problems. Polynomials are used to model many real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: What are some common mistakes to avoid when finding a polynomial with given zeros?

A: Some common mistakes to avoid when finding a polynomial with given zeros include:

• Not identifying the zeros of the polynomial
• Not multiplying the factors together correctly
• Not simplifying the polynomial correctly
• Not using the correct degree of the polynomial

Q: How do I check my work when finding a polynomial with given zeros?

A: To check your work when finding a polynomial with given zeros, you can use the following steps:

• Verify that the polynomial has the given zeros
• Check that the degree of the polynomial is correct
• Simplify the polynomial and verify that it is correct
• Use a calculator or computer algebra system to verify the polynomial

Q: What are some common resources for learning about polynomials?

A: Some common resources for learning about polynomials include:

• Textbooks on algebra and calculus
• Online resources, such as Khan Academy and Mathway
• Graphing calculators and computer algebra systems
• Online communities, such as Reddit's r/learnmath and r/math

Q: Can I use polynomials to solve systems of equations?

A: Yes, you can use polynomials to solve systems of equations. Polynomials are used to model many real-world phenomena, and solving systems of equations is an important part of many mathematical and scientific applications.

Q: What are some common applications of polynomials in science?

A: Polynomials have many common applications in science, including:

• Modeling the motion of objects
• Describing the growth of populations
• Analyzing the behavior of electrical circuits
• Solving systems of equations
• Finding the roots of a polynomial

Q: Can I use polynomials to model real-world phenomena?

A: Yes, you can use polynomials to model real-world phenomena. Polynomials are used to model many real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: What are some common mistakes to avoid when using polynomials to model real-world phenomena?

A: Some common mistakes to avoid when using polynomials to model real-world phenomena include:

• Not identifying the zeros of the polynomial
• Not multiplying the factors together correctly
• Not simplifying the polynomial correctly
• Not using the correct degree of the polynomial

Q: How do I check my work when using polynomials to model real-world phenomena?

A: To check your work when using polynomials to model real-world phenomena, you can use the following steps:

• Verify that the polynomial has the given zeros
• Check that the degree of the polynomial is correct
• Simplify the polynomial and verify that it is correct
• Use a calculator or computer algebra system to verify the polynomial

Q: What are some common resources for learning about polynomials in science?

A: Some common resources for learning about polynomials in science include:

• Textbooks on algebra and calculus
• Online resources, such as Khan Academy and Mathway
• Graphing calculators and computer algebra systems
• Online communities, such as Reddit's r/learnmath and r/math

Q: Can I use polynomials to solve optimization problems?

A: Yes, you can use polynomials to solve optimization problems. Polynomials are used to model many real-world phenomena, and solving optimization problems is an important part of many mathematical and scientific applications.

Q: What are some common applications of polynomials in optimization?

A: Polynomials have many common applications in optimization, including:

• Modeling the motion of objects
• Describing the growth of populations
• Analyzing the behavior of electrical circuits
• Solving systems of equations
• Finding the roots of a polynomial

Q: Can I use polynomials to model real-world optimization problems?

A: Yes, you can use polynomials to model real-world optimization problems. Polynomials are used to model many real-world phenomena, and solving optimization problems is an important part of many mathematical and scientific applications.

Q: What are some common mistakes to avoid when using polynomials to model real-world optimization problems?

A: Some common mistakes to avoid when using polynomials to model real-world optimization problems include:

• Not identifying the zeros of the polynomial
• Not multiplying the factors together correctly
• Not simplifying the polynomial correctly
• Not using the correct degree of the polynomial

Q: How do I check my work when using polynomials to model real-world optimization problems?

A: To check your work when using polynomials to model real-world optimization problems, you can use the following steps:

• Verify that the polynomial has the given zeros
• Check that the degree of the polynomial is correct
• Simplify the polynomial and verify that it is correct
• Use a calculator or computer algebra system to verify the polynomial

Q: What are some common resources for learning about polynomials in optimization?

A: Some common resources for learning about polynomials in optimization include:

• Textbooks on algebra and calculus
• Online resources, such as Khan Academy and Mathway
• Graphing calculators and computer algebra systems
• Online communities, such as Reddit's r/learnmath and r/math