Find The Quadratic Polynomial Each With The Given Numbers As The Sum And Product Of Its Zeros Respectively: (iii) 0,√5​

Overview of Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two, which means the highest power of the variable is two. It is in the form of ax^2 + bx + c, where a, b, and c are constants, and a cannot be zero. The zeros of a quadratic polynomial are the values of x that satisfy the equation when it is set to zero. In this article, we will focus on finding the quadratic polynomial with given sum and product of zeros.

Sum and Product of Zeros

The sum of the zeros of a quadratic polynomial is given by the formula -b/a, and the product of the zeros is given by the formula c/a. We are given the sum and product of the zeros as 0 and √5, respectively.

Finding the Quadratic Polynomial

To find the quadratic polynomial, we need to find the values of a, b, and c. We know that the sum of the zeros is -b/a, and the product of the zeros is c/a. We can use this information to find the values of a, b, and c.

Let's assume that the zeros are x and y. Then, we can write the following equations:

x + y = 0 ... (1) xy = √5 ... (2)

From equation (1), we can write y = -x. Substituting this value of y in equation (2), we get:

x(-x) = √5 -x^2 = √5 x^2 = -√5

Now, we can find the values of a, b, and c. We know that the sum of the zeros is -b/a, and the product of the zeros is c/a. We can write the following equations:

-b/a = 0 c/a = √5

From the first equation, we can write b = 0. From the second equation, we can write c = a√5.

Writing the Quadratic Polynomial

Now that we have found the values of a, b, and c, we can write the quadratic polynomial. We know that the quadratic polynomial is in the form of ax^2 + bx + c. Substituting the values of a, b, and c, we get:

a(x^2) + 0(x) + a√5 a(x^2 + √5)

Since a cannot be zero, we can assume that a = 1. Then, the quadratic polynomial becomes:

x^2 + √5

Conclusion

In this article, we found the quadratic polynomial with given sum and product of zeros as 0 and √5, respectively. We used the formulas for the sum and product of zeros to find the values of a, b, and c. We then wrote the quadratic polynomial in the form of ax^2 + bx + c. The final quadratic polynomial is x^2 + √5.

Example Problems

• Find the quadratic polynomial with given sum and product of zeros as 1 and 2, respectively.
• Find the quadratic polynomial with given sum and product of zeros as -1 and -3, respectively.

Solutions to Example Problems

• Let's assume that the zeros are x and y. Then, we can write the following equations:

x + y = 1 ... (1) xy = 2 ... (2)

From equation (1), we can write y = 1 - x. Substituting this value of y in equation (2), we get:

x(1 - x) = 2 -x^2 + x = 2 x^2 - x - 2 = 0

Now, we can find the values of a, b, and c. We know that the sum of the zeros is -b/a, and the product of the zeros is c/a. We can write the following equations:

-b/a = 1 c/a = 2

From the first equation, we can write b = -a. From the second equation, we can write c = 2a.

Now, we can write the quadratic polynomial. We know that the quadratic polynomial is in the form of ax^2 + bx + c. Substituting the values of a, b, and c, we get:

a(x^2) - a(x) + 2a a(x^2 - x + 2)

Since a cannot be zero, we can assume that a = 1. Then, the quadratic polynomial becomes:

x^2 - x + 2

• Let's assume that the zeros are x and y. Then, we can write the following equations:

x + y = -1 ... (1) xy = -3 ... (2)

From equation (1), we can write y = -1 - x. Substituting this value of y in equation (2), we get:

x(-1 - x) = -3 -x^2 - x = -3 x^2 + x + 3 = 0

Now, we can find the values of a, b, and c. We know that the sum of the zeros is -b/a, and the product of the zeros is c/a. We can write the following equations:

-b/a = -1 c/a = -3

From the first equation, we can write b = a. From the second equation, we can write c = -3a.

Now, we can write the quadratic polynomial. We know that the quadratic polynomial is in the form of ax^2 + bx + c. Substituting the values of a, b, and c, we get:

a(x^2) + a(x) - 3a a(x^2 + x - 3)

Since a cannot be zero, we can assume that a = 1. Then, the quadratic polynomial becomes:

x^2 + x - 3

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about quadratic polynomials.

Q: What is a quadratic polynomial?

A: A quadratic polynomial is a polynomial of degree two, which means the highest power of the variable is two. It is in the form of ax^2 + bx + c, where a, b, and c are constants, and a cannot be zero.

Q: What are the zeros of a quadratic polynomial?

A: The zeros of a quadratic polynomial are the values of x that satisfy the equation when it is set to zero. In other words, they are the values of x that make the polynomial equal to zero.

Q: How do I find the sum and product of the zeros of a quadratic polynomial?

A: The sum of the zeros of a quadratic polynomial is given by the formula -b/a, and the product of the zeros is given by the formula c/a.

Q: How do I find the quadratic polynomial with given sum and product of zeros?

A: To find the quadratic polynomial with given sum and product of zeros, you need to find the values of a, b, and c. You can use the formulas for the sum and product of zeros to find these values.

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

A: A quadratic polynomial is a polynomial of degree two, while a linear polynomial is a polynomial of degree one. In other words, a quadratic polynomial has a highest power of two, while a linear polynomial has a highest power of one.

Q: Can a quadratic polynomial have more than two zeros?

A: No, a quadratic polynomial can only have two zeros. This is because a quadratic polynomial is of degree two, and it can only have two roots.

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

A: To find the quadratic polynomial with given zeros, you need to find the values of a, b, and c. You can use the fact that the zeros are the values of x that satisfy the equation when it is set to zero.

Q: What is the relationship between the sum and product of the zeros of a quadratic polynomial?

A: The sum of the zeros of a quadratic polynomial is given by the formula -b/a, and the product of the zeros is given by the formula c/a. This means that the sum and product of the zeros are related to the coefficients of the quadratic polynomial.

Q: Can a quadratic polynomial have complex zeros?

A: Yes, a quadratic polynomial can have complex zeros. In fact, complex zeros are a common occurrence in quadratic polynomials.

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

A: To find the quadratic polynomial with given complex zeros, you need to find the values of a, b, and c. You can use the fact that the complex zeros are the values of x that satisfy the equation when it is set to zero.

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

A: A quadratic polynomial is a polynomial of degree two, while a cubic polynomial is a polynomial of degree three. In other words, a quadratic polynomial has a highest power of two, while a cubic polynomial has a highest power of three.

Q: Can a quadratic polynomial have more than two terms?

A: No, a quadratic polynomial can only have three terms: ax^2, bx, and c. This is because a quadratic polynomial is of degree two, and it can only have three terms.

Q: How do I find the quadratic polynomial with given terms?

A: To find the quadratic polynomial with given terms, you need to find the values of a, b, and c. You can use the fact that the terms are the values of x that satisfy the equation when it is set to zero.

Q: What is the relationship between the coefficients of a quadratic polynomial?

A: The coefficients of a quadratic polynomial are related to the sum and product of the zeros. In fact, the sum of the zeros is given by the formula -b/a, and the product of the zeros is given by the formula c/a.

Q: Can a quadratic polynomial have negative coefficients?

A: Yes, a quadratic polynomial can have negative coefficients. In fact, negative coefficients are a common occurrence in quadratic polynomials.

Q: How do I find the quadratic polynomial with given negative coefficients?

A: To find the quadratic polynomial with given negative coefficients, you need to find the values of a, b, and c. You can use the fact that the coefficients are the values of x that satisfy the equation when it is set to zero.

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

A: A quadratic polynomial is a polynomial of degree two, while a rational polynomial is a polynomial that can be expressed as the ratio of two polynomials. In other words, a quadratic polynomial has a highest power of two, while a rational polynomial has a highest power of one.

Q: Can a quadratic polynomial have more than two rational zeros?

A: No, a quadratic polynomial can only have two rational zeros. This is because a quadratic polynomial is of degree two, and it can only have two roots.

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

A: To find the quadratic polynomial with given rational zeros, you need to find the values of a, b, and c. You can use the fact that the rational zeros are the values of x that satisfy the equation when it is set to zero.

Q: What is the relationship between the rational zeros of a quadratic polynomial?

A: The rational zeros of a quadratic polynomial are related to the coefficients of the polynomial. In fact, the rational zeros are given by the formula x = -b/a.

Q: Can a quadratic polynomial have complex rational zeros?

A: Yes, a quadratic polynomial can have complex rational zeros. In fact, complex rational zeros are a common occurrence in quadratic polynomials.

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

A: To find the quadratic polynomial with given complex rational zeros, you need to find the values of a, b, and c. You can use the fact that the complex rational zeros are the values of x that satisfy the equation when it is set to zero.

Q: What is the difference between a quadratic polynomial and a polynomial with complex coefficients?

A: A quadratic polynomial is a polynomial of degree two with real coefficients, while a polynomial with complex coefficients is a polynomial of degree two with complex coefficients. In other words, a quadratic polynomial has real coefficients, while a polynomial with complex coefficients has complex coefficients.

Q: Can a quadratic polynomial have complex coefficients?

A: No, a quadratic polynomial cannot have complex coefficients. This is because a quadratic polynomial is of degree two, and it can only have real coefficients.

Q: How do I find the quadratic polynomial with given complex coefficients?

A: To find the quadratic polynomial with given complex coefficients, you need to find the values of a, b, and c. You can use the fact that the complex coefficients are the values of x that satisfy the equation when it is set to zero.

Q: What is the relationship between the complex coefficients of a quadratic polynomial?

A: The complex coefficients of a quadratic polynomial are related to the sum and product of the zeros. In fact, the sum of the zeros is given by the formula -b/a, and the product of the zeros is given by the formula c/a.

Q: Can a quadratic polynomial have negative complex coefficients?

A: Yes, a quadratic polynomial can have negative complex coefficients. In fact, negative complex coefficients are a common occurrence in quadratic polynomials.

Q: How do I find the quadratic polynomial with given negative complex coefficients?

A: To find the quadratic polynomial with given negative complex coefficients, you need to find the values of a, b, and c. You can use the fact that the negative complex coefficients are the values of x that satisfy the equation when it is set to zero.

Q: What is the difference between a quadratic polynomial and a polynomial with complex roots?

A: A quadratic polynomial is a polynomial of degree two with real coefficients, while a polynomial with complex roots is a polynomial of degree two with complex roots. In other words, a quadratic polynomial has real coefficients, while a polynomial with complex roots has complex roots.

Q: Can a quadratic polynomial have complex roots?

A: No, a quadratic polynomial cannot have complex roots. This is because a quadratic polynomial is of degree two, and it can only have real coefficients.

Q: How do I find the quadratic polynomial with given complex roots?

A: To find the quadratic polynomial with given complex roots, you need to find the values of a, b, and c. You can use the fact that the complex roots are the values of x that satisfy the equation when it is set to zero.

Q: What is the relationship between the complex roots of a quadratic polynomial?

A: The complex roots of a quadratic polynomial are related to the sum and product of the zeros. In fact, the sum of the zeros is given by the formula -b/a, and the product of the zeros is given by the formula c/a.

Q: Can a quadratic polynomial have negative complex roots?

A: Yes, a quadratic polynomial can have negative complex roots. In fact, negative complex roots are a common occurrence in quadratic polynomials.

Q: How do I find the quadratic polynomial with given negative complex roots?

A: To find the quadratic polynomial with given negative complex roots, you need to find the values of a, b, and c. You can use the fact that the negative complex roots are the values of x that satisfy the equation when it is set to zero.

Q: What is the difference between a quadratic polynomial and a polynomial with irrational roots?

A: A quadratic polynomial is a polynomial of degree two with real coefficients, while