The Relationship Between the Zeroes of Two Polynomials

Oct 13, 2025 by talynt.org 126 views

Introduction

In algebra, polynomials are mathematical expressions 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 explore the relationship between the zeroes of two polynomials and use this relationship to find the values of the coefficients of the first polynomial.

The Relationship Between the Zeroes of Two Polynomials

Let's consider two polynomials:

x^2 + px + q

2x^2 - 5x - 3

We are given that the zeroes of the first polynomial are double in value compared to the zeroes of the second polynomial. This means that if the zeroes of the second polynomial are $r_1$ and $r_2$ , then the zeroes of the first polynomial are $2r_1$ and $2r_2$ .

Finding the Zeroes of the Second Polynomial

To find the zeroes of the second polynomial, we can use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 2$ , $b = -5$ , and $c = -3$ . Plugging these values into the quadratic formula, we get:

x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}

x = \frac{5 \pm \sqrt{25 + 24}}{4}

x = \frac{5 \pm \sqrt{49}}{4}

x = \frac{5 \pm 7}{4}

So, the zeroes of the second polynomial are $r_1 = \frac{5 + 7}{4} = 3$ and $r_2 = \frac{5 - 7}{4} = -\frac{1}{2}$ .

Finding the Zeroes of the First Polynomial

Since the zeroes of the first polynomial are double in value compared to the zeroes of the second polynomial, we can write:

2r_1 = 2(3) = 6

2r_2 = 2\left(-\frac{1}{2}\right) = -1

So, the zeroes of the first polynomial are $6$ and $-1$ .

Finding the Values of p and q

Now that we have the zeroes of the first polynomial, we can use the fact that the product of the zeroes of a polynomial is equal to the constant term divided by the leading coefficient. In this case, the product of the zeroes is $6 \times (-1) = -6$ . Since the leading coefficient is $1$ , we can write:

q = -6

To find the value of $p$ , we can use the fact that the sum of the zeroes of a polynomial is equal to the negation of the coefficient of the linear term divided by the leading coefficient. In this case, the sum of the zeroes is $6 + (-1) = 5$ . Since the leading coefficient is $1$ , we can write:

p = -5

Conclusion

In this article, we explored the relationship between the zeroes of two polynomials and used this relationship to find the values of the coefficients of the first polynomial. We found that the zeroes of the first polynomial are double in value compared to the zeroes of the second polynomial, and we used this fact to find the values of $p$ and $q$ . The values of $p$ and $q$ are $-5$ and $-6$ , respectively.

References

[1] "Quadratic Formula". MathWorld.
[2] "Polynomial". MathWorld.

Introduction

In our previous article, we explored the relationship between the zeroes of two polynomials and used this relationship to find the values of the coefficients of the first polynomial. We found that the zeroes of the first polynomial are double in value compared to the zeroes of the second polynomial, and we used this fact to find the values of $p$ and $q$ . In this article, we will answer some frequently asked questions related to this topic.

Q: What is the relationship between the zeroes of two polynomials?

A: The zeroes of two polynomials are related in such a way that if the zeroes of the second polynomial are $r_1$ and $r_2$ , then the zeroes of the first polynomial are $2r_1$ and $2r_2$ .

Q: How do you find the zeroes of a polynomial?

A: To find the zeroes of a polynomial, you can use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the product of the zeroes of a polynomial?

A: The product of the zeroes of a polynomial is equal to the constant term divided by the leading coefficient.

Q: What is the sum of the zeroes of a polynomial?

A: The sum of the zeroes of a polynomial is equal to the negation of the coefficient of the linear term divided by the leading coefficient.

Q: How do you find the values of p and q in a polynomial?

A: To find the values of $p$ and $q$ in a polynomial, you can use the fact that the product of the zeroes is equal to the constant term divided by the leading coefficient, and the sum of the zeroes is equal to the negation of the coefficient of the linear term divided by the leading coefficient.

Q: What are the values of p and q in the polynomial $x^2 + px + q$ ?

A: The values of $p$ and $q$ in the polynomial $x^2 + px + q$ are $-5$ and $-6$ , respectively.

Q: Can you give an example of a polynomial where the zeroes are double in value compared to the zeroes of another polynomial?

A: Yes, consider the polynomials $x^2 + 2x + 4$ and $2x^2 - 5x - 3$ . The zeroes of the second polynomial are $3$ and $-\frac{1}{2}$ , and the zeroes of the first polynomial are $6$ and $-1$ , which are double in value compared to the zeroes of the second polynomial.

Q: How do you determine if the zeroes of a polynomial are double in value compared to the zeroes of another polynomial?

A: To determine if the zeroes of a polynomial are double in value compared to the zeroes of another polynomial, you can compare the values of the zeroes of the two polynomials.

Q: What are some real-world applications of the relationship between the zeroes of two polynomials?

A: The relationship between the zeroes of two polynomials has many real-world applications, such as in physics, engineering, and economics. For example, it can be used to model the motion of objects, the behavior of electrical circuits, and the growth of populations.

Conclusion

In this article, we answered some frequently asked questions related to the relationship between the zeroes of two polynomials and the values of the coefficients of the first polynomial. We hope that this article has been helpful in clarifying some of the concepts and providing a better understanding of the topic.

References

[1] "Quadratic Formula". MathWorld.
[2] "Polynomial". MathWorld.
[3] "Algebra". Khan Academy.
[4] "Polynomials". Khan Academy.