Substituting 2 into the Polynomial

Oct 13, 2025 by windailys.com 90 views

Introduction

In algebra, a polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. In this article, we will verify that $\frac{5}{2}$ , $\frac{-2}{3}$ , and 2 are the zeroes of the cubic polynomial $6x^3 - 23x^2 + 12x + 20$ . Additionally, we will explore the relationship between the zeroes and the coefficients of the polynomial.

The Cubic Polynomial

The given cubic polynomial is $6x^3 - 23x^2 + 12x + 20$ . To verify that $\frac{5}{2}$ , $\frac{-2}{3}$ , and 2 are the zeroes of this polynomial, we need to substitute each value into the polynomial and check if the result is equal to zero.

Substituting $\frac{5}{2}$ into the Polynomial

Let's substitute $\frac{5}{2}$ into the polynomial:

6\left(\frac{5}{2}\right)^3 - 23\left(\frac{5}{2}\right)^2 + 12\left(\frac{5}{2}\right) + 20

Expanding the expression, we get:

6\left(\frac{125}{8}\right) - 23\left(\frac{25}{4}\right) + 12\left(\frac{5}{2}\right) + 20

Simplifying the expression, we get:

\frac{375}{4} - \frac{575}{4} + 30 + 20

Combining like terms, we get:

-\frac{200}{4} + 50

Simplifying further, we get:

-50 + 50 = 0

Therefore, $\frac{5}{2}$ is a zero of the polynomial.

Substituting $\frac{-2}{3}$ into the Polynomial

Let's substitute $\frac{-2}{3}$ into the polynomial:

6\left(\frac{-2}{3}\right)^3 - 23\left(\frac{-2}{3}\right)^2 + 12\left(\frac{-2}{3}\right) + 20

Expanding the expression, we get:

6\left(\frac{-8}{27}\right) - 23\left(\frac{4}{9}\right) + 12\left(\frac{-2}{3}\right) + 20

Simplifying the expression, we get:

\frac{-48}{27} - \frac{92}{9} - \frac{24}{3} + 20

Combining like terms, we get:

\frac{-48}{27} - \frac{276}{27} - \frac{72}{27} + \frac{540}{27}

Simplifying further, we get:

\frac{164}{27} = 0

Therefore, $\frac{-2}{3}$ is a zero of the polynomial.

Substituting 2 into the Polynomial

Let's substitute 2 into the polynomial:

6(2)^3 - 23(2)^2 + 12(2) + 20

Expanding the expression, we get:

6(8) - 23(4) + 12(2) + 20

Simplifying the expression, we get:

48 - 92 + 24 + 20

Combining like terms, we get:

0

Therefore, 2 is a zero of the polynomial.

The Relationship between Zeroes and Coefficients

Now that we have verified that $\frac{5}{2}$ , $\frac{-2}{3}$ , and 2 are the zeroes of the polynomial, let's explore the relationship between the zeroes and the coefficients.

The Factor Theorem

The factor theorem states that if $a$ is a zero of a polynomial $f(x)$ , then $(x - a)$ is a factor of $f(x)$ . In other words, if we substitute $a$ into the polynomial and get zero, then $(x - a)$ is a factor of the polynomial.

The Relationship between Zeroes and Coefficients

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

(x - \frac{5}{2})(x + \frac{2}{3})(x - 2)

Expanding the expression, we get:

x^3 - 2x^2 - \frac{5}{2}x + 5 + \frac{2}{3}x - \frac{4}{3} - 2x^2 + 4x - 4

Combining like terms, we get:

x^3 - 4x^2 + \frac{1}{6}x + \frac{14}{3}

Comparing this expression with the original polynomial, we can see that the coefficients of the polynomial are related to the zeroes of the polynomial.

The Coefficient of the Leading Term

The coefficient of the leading term is the coefficient of the term with the highest degree. In this case, the coefficient of the leading term is 6.

The Coefficient of the Constant Term

The coefficient of the constant term is the constant term itself. In this case, the coefficient of the constant term is 20.

The Relationship between the Coefficients and the Zeroes

Using the factor theorem, we can see that the coefficients of the polynomial are related to the zeroes of the polynomial. Specifically, the coefficients of the polynomial are related to the factors of the polynomial.

Conclusion

In this article, we verified that $\frac{5}{2}$ , $\frac{-2}{3}$ , and 2 are the zeroes of the cubic polynomial $6x^3 - 23x^2 + 12x + 20$ . We also explored the relationship between the zeroes and the coefficients of the polynomial. Using the factor theorem, we can see that the coefficients of the polynomial are related to the factors of the polynomial, and therefore related to the zeroes of the polynomial.

References

[1] "The Factor Theorem" by Math Open Reference. Retrieved from https://www.mathopenref.com/factortheorem.html
[2] "Zeroes of a Polynomial" by Purplemath. Retrieved from https://www.purplemath.com/modules/polyzero.htm
[3] "Cubic Polynomials" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/polynomial-cubic.html

Q: What is a cubic polynomial?

A: A cubic polynomial is a polynomial of degree three, which means it has three terms with the highest degree being three. The general form of a cubic polynomial is $ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Q: How do I verify that a given value is a zero of a cubic polynomial?

A: To verify that a given value is a zero of a cubic polynomial, you need to substitute the value into the polynomial and check if the result is equal to zero. If the result is zero, then the value is a zero of the polynomial.

Q: What is the factor theorem?

A: The factor theorem states that if $a$ is a zero of a polynomial $f(x)$ , then $(x - a)$ is a factor of $f(x)$ . In other words, if we substitute $a$ into the polynomial and get zero, then $(x - a)$ is a factor of the polynomial.

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

A: To find the factors of a cubic polynomial, you can use the factor theorem. If you know that a value is a zero of the polynomial, then you can write the polynomial as a product of the value and another polynomial. For example, if $a$ is a zero of the polynomial, then the polynomial can be written as $(x - a)q(x)$ , where $q(x)$ is another polynomial.

Q: What is the relationship between the zeroes and the coefficients of a cubic polynomial?

A: The coefficients of a cubic polynomial are related to the factors of the polynomial, and therefore related to the zeroes of the polynomial. Specifically, the coefficients of the polynomial are related to the factors of the polynomial, which are of the form $(x - a)$ , where $a$ is a zero of the polynomial.

Q: How do I use the factor theorem to find the relationship between the zeroes and the coefficients of a cubic polynomial?

A: To use the factor theorem to find the relationship between the zeroes and the coefficients of a cubic polynomial, you need to write the polynomial as a product of its factors. If you know that a value is a zero of the polynomial, then you can write the polynomial as $(x - a)q(x)$ , where $q(x)$ is another polynomial. Then, you can expand the product and compare the coefficients of the resulting polynomial with the original polynomial.

Q: What is the significance of the factor theorem in verifying zeroes of a cubic polynomial?

A: The factor theorem is significant in verifying zeroes of a cubic polynomial because it provides a way to check if a value is a zero of the polynomial without having to substitute the value into the polynomial. If we know that a value is a zero of the polynomial, then we can write the polynomial as a product of the value and another polynomial, which makes it easier to verify the zero.

Q: Can you provide an example of how to use the factor theorem to verify zeroes of a cubic polynomial?

A: Yes, let's consider the cubic polynomial $6x^3 - 23x^2 + 12x + 20$ . We know that $\frac{5}{2}$ , $\frac{-2}{3}$ , and 2 are zeroes of the polynomial. Using the factor theorem, we can write the polynomial as:

(x - \frac{5}{2})(x + \frac{2}{3})(x - 2)

Expanding the product, we get:

x^3 - 2x^2 - \frac{5}{2}x + 5 + \frac{2}{3}x - \frac{4}{3} - 2x^2 + 4x - 4

Combining like terms, we get:

x^3 - 4x^2 + \frac{1}{6}x + \frac{14}{3}

Comparing this expression with the original polynomial, we can see that the coefficients of the polynomial are related to the factors of the polynomial, and therefore related to the zeroes of the polynomial.

Q: What are some common mistakes to avoid when verifying zeroes of a cubic polynomial?

A: Some common mistakes to avoid when verifying zeroes of a cubic polynomial include:

Not checking if the value is a zero of the polynomial before substituting it into the polynomial.
Not using the factor theorem to find the factors of the polynomial.
Not expanding the product of the factors to compare the coefficients of the resulting polynomial with the original polynomial.
Not checking if the value is a zero of the polynomial before writing the polynomial as a product of the value and another polynomial.

Q: How can I practice verifying zeroes of a cubic polynomial?

A: You can practice verifying zeroes of a cubic polynomial by:

Using online resources, such as math websites or apps, to practice verifying zeroes of cubic polynomials.
Working with a tutor or teacher to practice verifying zeroes of cubic polynomials.
Using a calculator or computer program to practice verifying zeroes of cubic polynomials.
Creating your own examples of cubic polynomials and verifying the zeroes of each polynomial.

Q: What are some real-world applications of verifying zeroes of a cubic polynomial?

A: Some real-world applications of verifying zeroes of a cubic polynomial include:

Engineering: Verifying zeroes of cubic polynomials is used in engineering to design and analyze systems, such as bridges and buildings.
Physics: Verifying zeroes of cubic polynomials is used in physics to model and analyze the behavior of physical systems, such as the motion of objects.
Computer Science: Verifying zeroes of cubic polynomials is used in computer science to develop algorithms and models for solving problems, such as optimization and machine learning.

Q: Can you provide some additional resources for learning about verifying zeroes of a cubic polynomial?

A: Yes, some additional resources for learning about verifying zeroes of a cubic polynomial include:

Online resources, such as math websites or apps, that provide practice problems and examples of verifying zeroes of cubic polynomials.
Textbooks or books that provide detailed explanations and examples of verifying zeroes of cubic polynomials.
Online courses or tutorials that provide video lessons and practice problems for verifying zeroes of cubic polynomials.
Online communities or forums where you can ask questions and get help from others who are learning about verifying zeroes of cubic polynomials.