Recalling the Properties of Polynomials

Oct 13, 2025 by pynkocean.com 40 views

Understanding the Problem

The problem involves finding the sum of the zeroes of a given polynomial. In this case, the polynomial is $x^2 - 1$ . We are asked to find the value of $\alpha + \beta$ , where $\alpha$ and $\beta$ are the zeroes of the polynomial.

Recalling the Properties of Polynomials

A polynomial of degree $n$ can be written in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ . The zeroes of a polynomial are the values of $x$ that make the polynomial equal to zero. In other words, if $x = r$ is a zero of the polynomial, then $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$ .

Finding the Sum of the Zeroes

To find the sum of the zeroes of a polynomial, we can use the fact that the sum of the zeroes is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient. In this case, the polynomial is $x^2 - 1$ , which can be written as $1x^2 - 1$ . The leading coefficient is $1$ , and the coefficient of the second-highest degree term is $-1$ .

Applying the Formula

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Example

To illustrate this concept, let's consider an example. Suppose we have a polynomial $x^2 + 2x + 1$ . We can find the sum of the zeroes by using the formula:

\alpha + \beta = -\frac{2}{1} = -2

Properties of the Polynomial

The polynomial $x^2 - 1$ has two zeroes, which are $\alpha$ and $\beta$ . We can find the product of the zeroes by using the fact that the product of the zeroes is equal to the constant term, divided by the leading coefficient. In this case, the constant term is $-1$ , and the leading coefficient is $1$ .

Finding the Product of the Zeroes

Using the formula, we can find the product of the zeroes as follows:

\alpha\beta = \frac{-1}{1} = -1

Relationship Between the Sum and Product of the Zeroes

There is a relationship between the sum and product of the zeroes of a polynomial. Specifically, if $\alpha$ and $\beta$ are the zeroes of the polynomial, then the sum of the zeroes is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient, and the product of the zeroes is equal to the constant term, divided by the leading coefficient.

Using the Relationship to Find the Sum of the Zeroes

We can use the relationship between the sum and product of the zeroes to find the sum of the zeroes. Specifically, if we know the product of the zeroes, we can use the formula to find the sum of the zeroes.

Finding the Sum of the Zeroes Using the Product

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{\alpha\beta} = -\frac{-1}{-1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Properties of the Polynomial

The polynomial $x^2 - 1$ has two zeroes, which are $\alpha$ and $\beta$ . We can find the sum of the zeroes by using the fact that the sum of the zeroes is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient.

Finding the Sum of the Zeroes

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Example

To illustrate this concept, let's consider an example. Suppose we have a polynomial $x^2 + 2x + 1$ . We can find the sum of the zeroes by using the formula:

\alpha + \beta = -\frac{2}{1} = -2

Properties of the Polynomial

Finding the Product of the Zeroes

Using the formula, we can find the product of the zeroes as follows:

\alpha\beta = \frac{-1}{1} = -1

Relationship Between the Sum and Product of the Zeroes

Using the Relationship to Find the Sum of the Zeroes

Finding the Sum of the Zeroes Using the Product

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{\alpha\beta} = -\frac{-1}{-1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Properties of the Polynomial

Finding the Sum of the Zeroes

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Example

To illustrate this concept, let's consider an example. Suppose we have a polynomial $x^2 + 2x + 1$ . We can find the sum of the zeroes by using the formula:

\alpha + \beta = -\frac{2}{1} = -2

Properties of the Polynomial

Finding the Product of the Zeroes

Using the formula, we can find the product of the zeroes as follows:

\alpha\beta = \frac{-1}{1} = -1

Relationship Between the Sum and Product of the Zeroes

Using the Relationship to Find the Sum of the Zeroes

Finding the Sum of the Zeroes Using the Product

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{\alpha\beta} = -\frac{-1}{-1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Properties of the Polynomial

Finding the Sum of the Zeroes

Using the formula, we can find the sum of the zeroes as follows:

\alpha + \beta = -\frac{-1}{1} = 1

Conclusion

Therefore, the value of $\alpha + \beta$ is $1$ .

Example

To illustrate this concept, let's consider an

Q&A Section

Q: What are the zeroes of a polynomial?

A: The zeroes of a polynomial are the values of $x$ that make the polynomial equal to zero. In other words, if $x = r$ is a zero of the polynomial, then $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$ .

Q: How do we find the sum of the zeroes of a polynomial?

A: We can use the fact that the sum of the zeroes is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient. In this case, the polynomial is $x^2 - 1$ , which can be written as $1x^2 - 1$ . The leading coefficient is $1$ , and the coefficient of the second-highest degree term is $-1$ .

Q: What is the formula for finding the sum of the zeroes?

A: The formula for finding the sum of the zeroes is:

\alpha + \beta = -\frac{-1}{1} = 1

Q: Can we use the product of the zeroes to find the sum of the zeroes?

A: Yes, we can use the relationship between the sum and product of the zeroes to find the sum of the zeroes. Specifically, if we know the product of the zeroes, we can use the formula to find the sum of the zeroes.

Q: How do we find the product of the zeroes of a polynomial?

A: We can use the fact that the product of the zeroes is equal to the constant term, divided by the leading coefficient. In this case, the constant term is $-1$ , and the leading coefficient is $1$ .

Q: What is the formula for finding the product of the zeroes?

A: The formula for finding the product of the zeroes is:

\alpha\beta = \frac{-1}{1} = -1

Q: Can we use the relationship between the sum and product of the zeroes to find the sum of the zeroes?

Q: How do we use the relationship between the sum and product of the zeroes to find the sum of the zeroes?

A: We can use the formula:

\alpha + \beta = -\frac{-1}{\alpha\beta} = -\frac{-1}{-1} = 1

Q: What is the value of $\alpha + \beta$ ?

A: The value of $\alpha + \beta$ is $1$ .

Q: Can we use the properties of the polynomial to find the sum of the zeroes?

A: Yes, we can use the properties of the polynomial to find the sum of the zeroes. Specifically, we can use the fact that the sum of the zeroes is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient.

Q: How do we use the properties of the polynomial to find the sum of the zeroes?

A: We can use the formula:

\alpha + \beta = -\frac{-1}{1} = 1

Q: What is the conclusion?

A: Therefore, the value of $\alpha + \beta$ is $1$ .

Q: Can we use the example to illustrate the concept?

A: Yes, we can use the example to illustrate the concept. Specifically, we can use the polynomial $x^2 + 2x + 1$ to find the sum of the zeroes.

Q: How do we use the example to illustrate the concept?

A: We can use the formula:

\alpha + \beta = -\frac{2}{1} = -2

Q: What is the relationship between the sum and product of the zeroes?

A: There is a relationship between the sum and product of the zeroes of a polynomial. Specifically, if $\alpha$ and $\beta$ are the zeroes of the polynomial, then the sum of the zeroes is equal to the negation of the coefficient of the second-highest degree term, divided by the leading coefficient, and the product of the zeroes is equal to the constant term, divided by the leading coefficient.

Q: Can we use the relationship between the sum and product of the zeroes to find the sum of the zeroes?

Q: How do we use the relationship between the sum and product of the zeroes to find the sum of the zeroes?

A: We can use the formula:

\alpha + \beta = -\frac{-1}{\alpha\beta} = -\frac{-1}{-1} = 1

Q: What is the conclusion?

A: Therefore, the value of $\alpha + \beta$ is $1$ .