If { \alpha, \beta$}$ Are Zeroes Of The Polynomial { P(x)=5x^2-6x+1$}$, Then Find The Value Of { \alpha+\beta+\alpha\beta$}$.
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 problem, we are given a quadratic polynomial {p(x)=5x^2-6x+1$}$ and we need to find the value of {\alpha+\beta+\alpha\beta$}$ where {\alpha$}$ and {\beta$}$ are the zeroes of the polynomial.
Properties of Quadratic Polynomials
A quadratic polynomial has the general form {ax^2+bx+c$}$, where {a$}$, {b$}$, and {c$}$ are constants. The zeroes of a quadratic polynomial can be found using the quadratic formula: {x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$. In this case, the quadratic polynomial is {p(x)=5x^2-6x+1$}$, so we have {a=5$}$, {b=-6$}$, and {c=1$}$.
Finding the Sum and Product of the Zeroes
The sum of the zeroes of a quadratic polynomial is given by the formula: {\alpha+\beta=-\fracb}{a}$}$. In this case, we have {\alpha+\beta=-\frac{-6}{5}=\frac{6}{5}$}$. The product of the zeroes of a quadratic polynomial is given by the formula{a}$}$. In this case, we have {\alpha\beta=\frac{1}{5}$}$.
Finding the Value of {\alpha+\beta+\alpha\beta$}$
Now that we have found the sum and product of the zeroes, we can find the value of {\alpha+\beta+\alpha\beta$}$ by simply adding the two values together: {\alpha+\beta+\alpha\beta=\frac{6}{5}+\frac{1}{5}=\frac{7}{5}$}$.
Conclusion
In this problem, we were given a quadratic polynomial and asked to find the value of {\alpha+\beta+\alpha\beta$}$ where {\alpha$}$ and {\beta$}$ are the zeroes of the polynomial. We used the properties of quadratic polynomials to find the sum and product of the zeroes, and then added the two values together to find the final answer.
Example
Let's consider an example to illustrate the concept. Suppose we have a quadratic polynomial {p(x)=2x^2-3x+1$}$. We can find the zeroes of the polynomial using the quadratic formula: {x=\frac-(-3)\pm\sqrt{(-3)^2-4(2)(1)}}{2(2)}$}$. Simplifying the expression, we get}4}=\frac{3\pm1}{4}$}$. Therefore, the zeroes of the polynomial are {x=1$}$ and {x=\frac{1}{2}$}$. We can now find the sum and product of the zeroes2}=\frac{3}{2}$}$ and {\alpha\beta=1\cdot\frac{1}{2}=\frac{1}{2}$}$. Finally, we can find the value of {\alpha+\beta+\alpha\beta$}$ by adding the two values together{2}+\frac{1}{2}=\frac{4}{2}=2$}$.
Applications
The concept of finding the sum and product of the zeroes of a quadratic polynomial has many applications in mathematics and science. For example, it is used in the study of quadratic equations, which are used to model a wide range of phenomena, including population growth, electrical circuits, and optics. It is also used in the study of quadratic forms, which are used to model the behavior of quadratic functions.
Future Research Directions
There are many open research directions in the study of quadratic polynomials and their zeroes. For example, researchers are still working to develop more efficient algorithms for finding the zeroes of quadratic polynomials, particularly for large polynomials. They are also working to develop new techniques for analyzing the behavior of quadratic functions, particularly in the context of quadratic forms.
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Linear Algebra and Its Applications" by Gilbert Strang
Keywords
- Quadratic polynomial
- Zeroes of a polynomial
- Sum and product of zeroes
- Quadratic formula
- Quadratic equations
- Quadratic forms
- Algebra
- Calculus
- Linear algebra
Summary
In this article, we discussed the concept of finding the sum and product of the zeroes of a quadratic polynomial. We used the properties of quadratic polynomials to find the sum and product of the zeroes, and then added the two values together to find the final answer. We also discussed the applications of this concept in mathematics and science, and identified some open research directions in the field.
Q&A
Q: What is a quadratic polynomial?
A: A quadratic polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It has the general form {ax^2+bx+c$}$, where {a$}$, {b$}$, and {c$}$ are constants.
Q: How do you find the zeroes of a quadratic polynomial?
A: The zeroes of a quadratic polynomial can be found using the quadratic formula: {x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$. This formula gives two solutions for the variable {x$}$, which are the zeroes of the polynomial.
Q: What is the sum of the zeroes of a quadratic polynomial?
A: The sum of the zeroes of a quadratic polynomial is given by the formula: {\alpha+\beta=-\frac{b}{a}$}$. This formula gives the sum of the two zeroes of the polynomial.
Q: What is the product of the zeroes of a quadratic polynomial?
A: The product of the zeroes of a quadratic polynomial is given by the formula: {\alpha\beta=\frac{c}{a}$}$. This formula gives the product of the two zeroes of the polynomial.
Q: How do you find the value of {\alpha+\beta+\alpha\beta$}$?
A: To find the value of {\alpha+\beta+\alpha\beta$}$, you need to find the sum and product of the zeroes of the polynomial, and then add the two values together: {\alpha+\beta+\alpha\beta=\frac{6}{5}+\frac{1}{5}=\frac{7}{5}$}$.
Q: What are some applications of finding the sum and product of the zeroes of a quadratic polynomial?
A: The concept of finding the sum and product of the zeroes of a quadratic polynomial has many applications in mathematics and science. For example, it is used in the study of quadratic equations, which are used to model a wide range of phenomena, including population growth, electrical circuits, and optics. It is also used in the study of quadratic forms, which are used to model the behavior of quadratic functions.
Q: What are some open research directions in the study of quadratic polynomials and their zeroes?
A: There are many open research directions in the study of quadratic polynomials and their zeroes. For example, researchers are still working to develop more efficient algorithms for finding the zeroes of quadratic polynomials, particularly for large polynomials. They are also working to develop new techniques for analyzing the behavior of quadratic functions, particularly in the context of quadratic forms.
Q: What are some common mistakes to avoid when finding the sum and product of the zeroes of a quadratic polynomial?
A: Some common mistakes to avoid when finding the sum and product of the zeroes of a quadratic polynomial include:
- Not using the correct formula for the sum and product of the zeroes
- Not simplifying the expression correctly
- Not checking the solutions for the variable {x$}$
- Not using the correct values for the coefficients {a$}$, {b$}$, and {c$}$
Q: How can I practice finding the sum and product of the zeroes of a quadratic polynomial?
A: You can practice finding the sum and product of the zeroes of a quadratic polynomial by working through examples and exercises. You can also try using online resources, such as calculators and software, to help you with the calculations.
Q: What are some tips for finding the sum and product of the zeroes of a quadratic polynomial?
A: Some tips for finding the sum and product of the zeroes of a quadratic polynomial include:
- Make sure to use the correct formula for the sum and product of the zeroes
- Simplify the expression correctly
- Check the solutions for the variable {x$}$
- Use the correct values for the coefficients {a$}$, {b$}$, and {c$}$
- Practice, practice, practice!
Keywords
- Quadratic polynomial
- Zeroes of a polynomial
- Sum and product of zeroes
- Quadratic formula
- Quadratic equations
- Quadratic forms
- Algebra
- Calculus
- Linear algebra
Summary
In this article, we discussed the concept of finding the sum and product of the zeroes of a quadratic polynomial. We used the properties of quadratic polynomials to find the sum and product of the zeroes, and then added the two values together to find the final answer. We also discussed the applications of this concept in mathematics and science, and identified some open research directions in the field.