Understanding Zeros and Degree

Oct 13, 2025 by talynt.org 94 views

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Introduction

In algebra, a polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we are given the zeros and degree of a polynomial, we can form the polynomial using this information. In this article, we will discuss how to form a polynomial whose zeros and degree are given.

Understanding Zeros and Degree

Before we proceed, let's understand the concepts of zeros and degree.

Zeros: The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a zero into the polynomial, the result will be zero.
Degree: The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $ax^2 + bx + c$ , the degree is 2.

Forming a Polynomial with Given Zeros

Now that we understand the concepts of zeros and degree, let's form a polynomial with given zeros.

Given zeros: $-3, 3, 7$ Degree: 3

We know that the polynomial will have the form $a(x + 3)(x - 3)(x - 7)$ , where $a$ is a constant.

Finding the Value of a

To find the value of $a$ , we can use the fact that the coefficient of the highest power of $x$ in the polynomial is equal to the product of the zeros.

In this case, the highest power of $x$ is $x^3$ , and the coefficient is $a$ . The product of the zeros is $(-3)(3)(7) = -63$ .

Therefore, we have $a = -63$ .

Writing the Polynomial

Now that we have found the value of $a$ , we can write the polynomial.

The polynomial is $-63(x + 3)(x - 3)(x - 7)$ .

Simplifying the Polynomial

To simplify the polynomial, we can multiply out the factors.

$-63(x + 3)(x - 3)(x - 7) = -63(x^2 - 9)(x - 7)$

$= -63(x^3 - 16x^2 + 63x - 63)$

$= -63x^3 + 1008x^2 - 3975x + 3969$

Conclusion

In this article, we have discussed how to form a polynomial whose zeros and degree are given. We have used the fact that the polynomial will have the form $a(x + 3)(x - 3)(x - 7)$ , where $a$ is a constant. We have found the value of $a$ using the fact that the coefficient of the highest power of $x$ in the polynomial is equal to the product of the zeros. Finally, we have written and simplified the polynomial.

Example Problems

Problem 1

Form a polynomial whose zeros are $-2, 4, 6$ and degree is 3.

Solution

The polynomial will have the form $a(x + 2)(x - 4)(x - 6)$ , where $a$ is a constant. To find the value of $a$ , we can use the fact that the coefficient of the highest power of $x$ in the polynomial is equal to the product of the zeros. In this case, the highest power of $x$ is $x^3$ , and the coefficient is $a$ . The product of the zeros is $(-2)(4)(6) = -48$ . Therefore, we have $a = -48$ . The polynomial is $-48(x + 2)(x - 4)(x - 6)$ .

Problem 2

Form a polynomial whose zeros are $-1, 2, 5$ and degree is 3.

Solution

The polynomial will have the form $a(x + 1)(x - 2)(x - 5)$ , where $a$ is a constant. To find the value of $a$ , we can use the fact that the coefficient of the highest power of $x$ in the polynomial is equal to the product of the zeros. In this case, the highest power of $x$ is $x^3$ , and the coefficient is $a$ . The product of the zeros is $(-1)(2)(5) = -10$ . Therefore, we have $a = -10$ . The polynomial is $-10(x + 1)(x - 2)(x - 5)$ .

Applications

Forming a polynomial with given zeros has many applications in mathematics and science. Some of the applications include:

Graphing: Forming a polynomial with given zeros allows us to graph the polynomial.
Solving Equations: Forming a polynomial with given zeros allows us to solve equations involving the polynomial.
Optimization: Forming a polynomial with given zeros allows us to optimize functions involving the polynomial.

Conclusion

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Frequently Asked Questions

Q1: What is the formula for forming a polynomial with given zeros?

A1: The formula for forming a polynomial with given zeros is $a(x - r_1)(x - r_2)...(x - r_n)$ , where $a$ is a constant and $r_1, r_2,...,r_n$ are the given zeros.

Q2: How do I find the value of a in the polynomial?

A2: To find the value of $a$ , you can use the fact that the coefficient of the highest power of $x$ in the polynomial is equal to the product of the zeros.

Q3: What is the degree of a polynomial with given zeros?

A3: The degree of a polynomial with given zeros is equal to the number of zeros.

Q4: Can I have a polynomial with more than one zero?

A4: Yes, you can have a polynomial with more than one zero. For example, the polynomial $x(x - 2)(x - 3)$ has three zeros: $0, 2,$ and $3$ .

Q5: How do I simplify a polynomial with given zeros?

A5: To simplify a polynomial with given zeros, you can multiply out the factors.

Q6: Can I have a polynomial with complex zeros?

A6: Yes, you can have a polynomial with complex zeros. For example, the polynomial $x(x - 2i)(x + 2i)$ has complex zeros: $0, 2i,$ and $-2i$ .

Q7: How do I find the zeros of a polynomial?

A7: To find the zeros of a polynomial, you can set the polynomial equal to zero and solve for $x$ .

Q8: Can I have a polynomial with rational zeros?

A8: Yes, you can have a polynomial with rational zeros. For example, the polynomial $x(x - 1)(x - 2)$ has rational zeros: $0, 1,$ and $2$ .

Q9: How do I find the degree of a polynomial with rational zeros?

A9: To find the degree of a polynomial with rational zeros, you can count the number of zeros.

Q10: Can I have a polynomial with irrational zeros?

A10: Yes, you can have a polynomial with irrational zeros. For example, the polynomial $x(x - \sqrt{2})(x + \sqrt{2})$ has irrational zeros: $0, \sqrt{2},$ and $-\sqrt{2}$ .

Example Problems

Problem 1

Find the zeros of the polynomial $x(x - 2)(x - 3)$ .

Solution

To find the zeros of the polynomial, we can set the polynomial equal to zero and solve for $x$ . We have $x(x - 2)(x - 3) = 0$ . Therefore, the zeros of the polynomial are $0, 2,$ and $3$ .

Problem 2

Find the degree of the polynomial $x(x - 2i)(x + 2i)$ .

Solution

To find the degree of the polynomial, we can count the number of zeros. We have three zeros: $0, 2i,$ and $-2i$ . Therefore, the degree of the polynomial is $3$ .

Problem 3

Find the value of $a$ in the polynomial $a(x - 1)(x - 2)$ .

Solution

To find the value of $a$ , we can use the fact that the coefficient of the highest power of $x$ in the polynomial is equal to the product of the zeros. We have $a(x - 1)(x - 2) = ax^2 - 3ax + 2a$ . Therefore, the coefficient of the highest power of $x$ is $a$ . The product of the zeros is $(-1)(-2) = 2$ . Therefore, we have $a = 2$ .

Conclusion

In this article, we have discussed some frequently asked questions about forming a polynomial with given zeros. We have provided answers to these questions and have also included some example problems to help illustrate the concepts. We hope that this article has been helpful in understanding how to form a polynomial with given zeros.