Find A Degree 3 Polynomial Having Zeros -8, 4, And 6, With The Coefficient Of $x^3$ Equal To 1.The Polynomial Is: $ P(x) = (x + 8)(x - 4)(x - 6) $

Introduction

In algebra, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified based on their degree, which is the highest power of the variable in the polynomial. A degree 3 polynomial, also known as a cubic polynomial, has the highest power of 3. In this article, we will discuss how to find a degree 3 polynomial with given zeros.

What are Zeros of a Polynomial?

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a zero of a polynomial into the polynomial, the result will be zero. For example, if we have a polynomial $ P(x) = x^2 - 4 $, the zeros of this polynomial are $ x = 2 $ and $ x = -2 $, because when we substitute $ x = 2 $ or $ x = -2 $ into the polynomial, we get $ P(2) = 2^2 - 4 = 0 $ and $ P(-2) = (-2)^2 - 4 = 0 $.

Finding a Degree 3 Polynomial with Given Zeros

To find a degree 3 polynomial with given zeros, we can use the fact that if $ r $ is a zero of a polynomial, then $ (x - r) $ is a factor of the polynomial. This means that if we know the zeros of a polynomial, we can write the polynomial as a product of linear factors.

For example, let's say we want to find a degree 3 polynomial with zeros $ -8, 4, $ and $ 6 $. We can write the polynomial as:

$$P(x) = (x + 8)(x - 4)(x - 6)$$

This polynomial has the desired zeros, because when we substitute $ x = -8, 4, $ or $ 6 $ into the polynomial, we get:

$$P(-8) = (-8 + 8)(-8 - 4)(-8 - 6) = 0 \cdot (-12) \cdot (-14) = 0$$

$$P(4) = (4 + 8)(4 - 4)(4 - 6) = 12 \cdot 0 \cdot (-2) = 0$$

$$P(6) = (6 + 8)(6 - 4)(6 - 6) = 14 \cdot 2 \cdot 0 = 0$$

Why is the Coefficient of $x^3$ Equal to 1?

In the polynomial $ P(x) = (x + 8)(x - 4)(x - 6) $, the coefficient of $ x^3 $ is equal to 1. This is because when we multiply the three linear factors together, the term with the highest power of $ x $ is $ x^3 $, and the coefficient of this term is 1.

To see why this is the case, let's expand the polynomial using the distributive property:

$$P(x) = (x + 8)(x - 4)(x - 6)$$

$$= (x^2 - 4x + 8x - 32)(x - 6)$$

$$= (x^2 + 4x - 32)(x - 6)$$

$$= x^3 - 6x^2 + 4x^2 - 24x - 32x + 192$$

$$= x^3 - 2x^2 - 56x + 192$$

As we can see, the coefficient of $ x^3 $ is indeed 1.

Conclusion

In this article, we discussed how to find a degree 3 polynomial with given zeros. We showed that if we know the zeros of a polynomial, we can write the polynomial as a product of linear factors. We also explained why the coefficient of $ x^3 $ is equal to 1 in the polynomial $ P(x) = (x + 8)(x - 4)(x - 6) $. We hope that this article has provided a clear understanding of how to find a degree 3 polynomial with given zeros.

Applications of Degree 3 Polynomials

Degree 3 polynomials have many applications in mathematics and science. For example, they can be used to model the motion of objects under the influence of gravity, or to describe the growth of populations over time. They can also be used to solve problems in engineering, economics, and other fields.

Solving Equations with Degree 3 Polynomials

Solving equations with degree 3 polynomials can be challenging, but there are several methods that can be used. One method is to use the quadratic formula, which can be used to solve quadratic equations. Another method is to use the rational root theorem, which can be used to find the rational roots of a polynomial.

Graphing Degree 3 Polynomials

Graphing degree 3 polynomials can be done using a variety of methods, including graphing calculators and computer software. The graph of a degree 3 polynomial can be used to visualize the behavior of the polynomial, and to identify its zeros and other important features.

Real-World Applications of Degree 3 Polynomials

Degree 3 polynomials have many real-world applications. For example, they can be used to model the motion of objects under the influence of gravity, or to describe the growth of populations over time. They can also be used to solve problems in engineering, economics, and other fields.

Conclusion

In conclusion, degree 3 polynomials are an important topic in mathematics and science. They have many applications, including modeling the motion of objects under the influence of gravity, and solving problems in engineering, economics, and other fields. We hope that this article has provided a clear understanding of how to find a degree 3 polynomial with given zeros, and has shown the importance of degree 3 polynomials in real-world applications.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Polynomials" by Wolfram MathWorld

Further Reading

For further reading on degree 3 polynomials, we recommend the following resources:

• [1] "Degree 3 Polynomials" by Math Open Reference
• [2] "Polynomial Equations" by Wolfram MathWorld
• [3] "Algebraic Curves" by Wikipedia

We hope that this article has provided a clear understanding of how to find a degree 3 polynomial with given zeros, and has shown the importance of degree 3 polynomials in real-world applications.

Q: What is a degree 3 polynomial?

A: A degree 3 polynomial, also known as a cubic polynomial, is a polynomial with the highest power of 3. It is a polynomial of the form $ ax^3 + bx^2 + cx + d $, where $ a, b, c, $ and $ d $ are constants, and $ a $ is not equal to 0.

Q: How do I find a degree 3 polynomial with given zeros?

A: To find a degree 3 polynomial with given zeros, you can use the fact that if $ r $ is a zero of a polynomial, then $ (x - r) $ is a factor of the polynomial. This means that if you know the zeros of a polynomial, you can write the polynomial as a product of linear factors.

Q: What are the zeros of a degree 3 polynomial?

A: The zeros of a degree 3 polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a zero of a polynomial into the polynomial, the result will be zero.

Q: How do I find the zeros of a degree 3 polynomial?

A: To find the zeros of a degree 3 polynomial, you can use the fact that if $ r $ is a zero of a polynomial, then $ (x - r) $ is a factor of the polynomial. This means that if you know the factors of a polynomial, you can find the zeros of the polynomial by setting each factor equal to zero and solving for $ x $.

Q: What is the coefficient of $x^3$ in a degree 3 polynomial?

A: The coefficient of $ x^3 $ in a degree 3 polynomial is the constant that multiplies the term $ x^3 $. In a degree 3 polynomial of the form $ ax^3 + bx^2 + cx + d $, the coefficient of $ x^3 $ is $ a $.

Q: How do I graph a degree 3 polynomial?

A: To graph a degree 3 polynomial, you can use a variety of methods, including graphing calculators and computer software. The graph of a degree 3 polynomial can be used to visualize the behavior of the polynomial, and to identify its zeros and other important features.

Q: What are some real-world applications of degree 3 polynomials?

A: Degree 3 polynomials have many real-world applications, including modeling the motion of objects under the influence of gravity, and solving problems in engineering, economics, and other fields.

Q: How do I solve equations with degree 3 polynomials?

A: Solving equations with degree 3 polynomials can be challenging, but there are several methods that can be used. One method is to use the quadratic formula, which can be used to solve quadratic equations. Another method is to use the rational root theorem, which can be used to find the rational roots of a polynomial.

Q: What is the difference between a degree 3 polynomial and a quadratic polynomial?

A: A degree 3 polynomial is a polynomial with the highest power of 3, while a quadratic polynomial is a polynomial with the highest power of 2. In other words, a degree 3 polynomial is a cubic polynomial, while a quadratic polynomial is a quadratic polynomial.

Q: Can I use a degree 3 polynomial to model a real-world situation?

A: Yes, you can use a degree 3 polynomial to model a real-world situation. For example, you can use a degree 3 polynomial to model the motion of an object under the influence of gravity, or to describe the growth of a population over time.

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

A: To determine the degree of a polynomial, you can look at the highest power of the variable in the polynomial. If the highest power is 3, then the polynomial is a degree 3 polynomial.

Q: What is the significance of the zeros of a degree 3 polynomial?

A: The zeros of a degree 3 polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a zero of a polynomial into the polynomial, the result will be zero. The zeros of a polynomial are important because they can be used to solve equations and to graph the polynomial.

Q: Can I use a degree 3 polynomial to solve a system of equations?

A: Yes, you can use a degree 3 polynomial to solve a system of equations. For example, you can use a degree 3 polynomial to solve a system of linear equations, or to solve a system of nonlinear equations.

Q: How do I find the roots of a degree 3 polynomial?

A: To find the roots of a degree 3 polynomial, you can use a variety of methods, including factoring, the quadratic formula, and the rational root theorem.

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

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

Q: Can I use a degree 3 polynomial to model a periodic function?

A: Yes, you can use a degree 3 polynomial to model a periodic function. For example, you can use a degree 3 polynomial to model the motion of a pendulum, or to describe the behavior of a population over time.

Q: How do I determine the number of zeros of a degree 3 polynomial?

A: To determine the number of zeros of a degree 3 polynomial, you can use the fact that a degree 3 polynomial can have at most 3 zeros. You can also use the rational root theorem to find the rational roots of the polynomial, and then use the quadratic formula to find the remaining roots.

Q: What is the significance of the coefficient of $x^3$ in a degree 3 polynomial?

A: The coefficient of $ x^3 $ in a degree 3 polynomial is the constant that multiplies the term $ x^3 $. In a degree 3 polynomial of the form $ ax^3 + bx^2 + cx + d $, the coefficient of $ x^3 $ is $ a $. The coefficient of $ x^3 $ is important because it determines the behavior of the polynomial as $ x $ approaches infinity.

Q: Can I use a degree 3 polynomial to model a function that has a discontinuity?

A: Yes, you can use a degree 3 polynomial to model a function that has a discontinuity. For example, you can use a degree 3 polynomial to model the behavior of a population over time, or to describe the motion of an object under the influence of gravity.

Q: How do I find the derivative of a degree 3 polynomial?

A: To find the derivative of a degree 3 polynomial, you can use the power rule of differentiation. The power rule states that if $ f(x) = x^n $, then $ f'(x) = nx^{n-1} $. You can also use the product rule and the quotient rule to find the derivative of a degree 3 polynomial.

Q: What is the difference between a degree 3 polynomial and a degree 2 polynomial?

A: A degree 3 polynomial is a polynomial with the highest power of 3, while a degree 2 polynomial is a polynomial with the highest power of 2. In other words, a degree 3 polynomial is a cubic polynomial, while a degree 2 polynomial is a quadratic polynomial.

Q: Can I use a degree 3 polynomial to model a function that has a periodic behavior?

A: Yes, you can use a degree 3 polynomial to model a function that has a periodic behavior. For example, you can use a degree 3 polynomial to model the motion of a pendulum, or to describe the behavior of a population over time.

Q: How do I find the integral of a degree 3 polynomial?

A: To find the integral of a degree 3 polynomial, you can use the power rule of integration. The power rule states that if $ f(x) = x^n $, then $ \int f(x) dx = \frac{x^{n+1}}{n+1} + C $. You can also use the substitution method and the integration by parts method to find the integral of a degree 3 polynomial.

Q: What is the significance of the zeros of a degree 3 polynomial in a real-world application?

A: The zeros of a degree 3 polynomial are the values of the variable that make the polynomial equal to zero. In a real-world application, the zeros of a polynomial can be used to model the behavior of a system, or to describe the growth of a population over time.

Q: Can I use a degree 3 polynomial to model a function that has a discontinuity in its derivative?

A: Yes, you can use a degree 3 polynomial to model a function that has a discontinuity in its derivative. For example, you can use a degree 3 polynomial to model the behavior of a population over time, or to describe the motion of an object under the influence of gravity.

Q: How do I determine the number of zeros of a degree 3 polynomial in a real-world application?

A: To determine the number of zeros of a degree 3 polynomial in a real-world application, you can use the fact that a degree