How To Form A Polynomial With Given Zeros And Degree?

How to Form a Polynomial with Given Zeros and Degree

Polynomials are an important part of mathematics, with applications in a wide variety of fields, from engineering to physics to economics. In this article, we will discuss how to form a polynomial with given zeros and degree. We will start by defining what a polynomial is and then discuss the relationship between the zeros and degree of a polynomial. We will then provide a step-by-step guide on how to form a polynomial with given zeros and degree.

By the end of this article, you will be able to:

• Define a polynomial
• Explain the relationship between the zeros and degree of a polynomial
• Form a polynomial with given zeros and degree

So let’s get started!

How To Form A Polynomial With Given Zeros And Degree?

| Zeros | Degree | Polynomial |
|—|—|—|
| -2, 3, 5 | 3 | $p(x) = x^3 – 14x^2 + 51x – 60$ |
| -1, 0, 2 | 3 | $p(x) = x^3 – 3x^2 + 4x – 2$ |
| 1, -2, -3 | 3 | $p(x) = x^3 – 6x^2 + 11x – 6$ |
| -1, $\frac{1}{2}$, $\frac{3}{2}$ | 3 | $p(x) = x^3 – \frac{5}{2}x^2 + \frac{19}{4}x – \frac{15}{8}$ |

In this tutorial, we will learn how to form a polynomial with given zeros and degree. We will start by discussing the Fundamental Theorem of Algebra, which states that a polynomial of degree n has n roots (counting multiplicities). We will then show how to find the complex roots of a polynomial using the quadratic formula, the cubic formula, or other methods. Finally, we will show how to find the real roots of a polynomial using the Descartes’ rule of signs or other methods.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree n has n roots (counting multiplicities). This means that if we have a polynomial of degree n, then we can find n complex numbers such that the polynomial is equal to zero when we plug these numbers into it.

For example, the polynomial $x^2 – 2x + 1$ has two roots, $1$ and $-1$. We can verify this by plugging these numbers into the polynomial:

$$
\begin{align*}
(1)^2 – 2(1) + 1 &= 0 \\
(-1)^2 – 2(-1) + 1 &= 0
\end{align*}
$$

The complex roots of a polynomial can be found using the quadratic formula, the cubic formula, or other methods. We will not discuss these methods in detail in this tutorial, but you can find more information about them in other resources.

Finding the Real Roots of a Polynomial

The real roots of a polynomial can be found using the Descartes’ rule of signs or other methods. The Descartes’ rule of signs states that the number of positive real roots of a polynomial is equal to the number of sign changes in the polynomial’s coefficients. For example, the polynomial $x^2 – 2x + 1$ has one positive real root because there is one sign change in the coefficients.

The Descartes’ rule of signs can be used to find the approximate location of the real roots of a polynomial. However, it cannot be used to find the exact location of the roots. To find the exact location of the roots, you can use a graphing calculator or other mathematical software.

Forming a Polynomial with Given Zeros

Once you have found the zeros of a polynomial, you can form the polynomial by multiplying together the linear factors corresponding to each zero. For example, if the polynomial has zeros $a$, $b$, and $c$, then the polynomial can be written as

$$
P(x) = (x – a)(x – b)(x – c)
$$

In this tutorial, we have learned how to form a polynomial with given zeros and degree. We have also discussed the Fundamental Theorem of Algebra and the methods for finding the real roots of a polynomial.

We hope that this tutorial has been helpful. If you have any questions, please feel free to ask in the comments below.

How To Form A Polynomial With Given Zeros And Degree?

A polynomial is a mathematical expression that is made up of a sum of terms, each of which is a product of a constant and a variable raised to a power. The degree of a polynomial is the highest power of the variable that appears in the polynomial.

Given a set of zeros of a polynomial, it is possible to find a unique polynomial with those zeros. This polynomial is called the Lagrange interpolation polynomial. The Lagrange interpolation polynomial is a polynomial of degree one less than the number of zeros.

To find the Lagrange interpolation polynomial, we first need to find the Lagrange basis polynomials. The Lagrange basis polynomials are a set of polynomials, one for each zero of the polynomial, that are defined as follows:

$$L_i(x) = \prod_{j \neq i} \frac{x – x_j}{x_i – x_j}$$

where $x_i$ is the $i$th zero of the polynomial.

Once we have the Lagrange basis polynomials, we can find the Lagrange interpolation polynomial by multiplying the Lagrange basis polynomials together and evaluating the product at $x = 0$.

The Lagrange interpolation polynomial is a very useful tool for finding polynomials with given zeros. It can be used to approximate functions, to solve polynomial equations, and to perform other mathematical operations.

Examples

Example 1: Form a polynomial with zeros 1, -2, and 3.

The first step is to find the Lagrange basis polynomials. We have three zeros, so we will need three Lagrange basis polynomials.

$$L_1(x) = \frac{x – (-2)}{1 – (-2)} = \frac{x + 2}{3}$$

$$L_2(x) = \frac{x – 3}{1 – 3} = \frac{x – 3}{-2} = -\frac{x + 3}{2}$$

$$L_3(x) = \frac{x – 1}{1 – 1} = x$$

Now we can multiply the Lagrange basis polynomials together and evaluate the product at $x = 0$.

$$P(x) = L_1(x)L_2(x)L_3(x) = \left(\frac{x + 2}{3}\right)\left(-\frac{x + 3}{2}\right)x = -\frac{x^3 + 7x^2 + 12x + 18}{6}$$

Therefore, the polynomial with zeros 1, -2, and 3 is $P(x) = -\frac{x^3 + 7x^2 + 12x + 18}{6}$.

Example 2: Form a polynomial with zeros -1, 2i, and -2i.

The first step is to find the Lagrange basis polynomials. We have three zeros, so we will need three Lagrange basis polynomials.

$$L_1(x) = \frac{x – (2i)}{1 – (2i)} = \frac{x – 2i}{-3i} = -\frac{x + 2i}{3i}$$

$$L_2(x) = \frac{x – (-2i)}{1 – (-2i)} = \frac{x + 2i}{3i}$$

$$L_3(x) = \frac{x – (-1)}{1 – (-1)} = x + 1$$

Now we can multiply the Lagrange basis polynomials together and evaluate the product at $x = 0$.

$$P(x) = L_1(x)L_2(x)L_3(x) = \left(-\frac{x + 2i}{3i}\right)\left(\frac{x + 2i}{3i}\right)(x + 1) = (x^2 + 2x + 1)$$

Therefore, the polynomial with zeros -1, 2i, and -2i is $P(x) = (x^2 + 2x + 1)$.

Example 3: Form a polynomial with zeros 0, 1, and -1.

The first step is to find the Lagrange basis polynomials. We have three zeros, so we will need three Lagrange basis polynomials.

$$L_0(x) = \frac{x

How to Form a Polynomial With Given Zeros and Degree?

Given a set of zeros of a polynomial, it is possible to find a unique polynomial with those zeros. The following steps show how to do this:

1. Find the leading coefficient. The leading coefficient is the coefficient of the term with the highest degree. In the polynomial $f(x) = ax^3 + bx^2 + cx + d$, the leading coefficient is $a$.
2. Find the constant term. The constant term is the term with the degree 0. In the polynomial $f(x) = ax^3 + bx^2 + cx + d$, the constant term is $d$.
3. Write the polynomial in factored form. The factored form of a polynomial is a product of linear factors. For example, the polynomial $f(x) = x^3 – 3x^2 + 4x – 12$ can be written in factored form as $f(x) = (x – 2)(x – 3)(x + 2)$.
4. Substitute the zeros into the factored form. Substitute each zero of the polynomial into the factored form. This will give you a set of equations that you can solve to find the coefficients of the polynomial.

For example, consider the polynomial $f(x) = x^3 – 3x^2 + 4x – 12$. The zeros of this polynomial are $2$, $3$, and $-2$. Substituting these zeros into the factored form gives us the following equations:

$$
(x – 2)(x – 3)(x + 2) = 0 \\
(x – 2)(x – 3) = 0 \\
x – 2 = 0 \\
x = 2 \\
(x – 3) = 0 \\
x = 3 \\
(x + 2) = 0 \\
x = -2
$$

Solving these equations gives us the coefficients of the polynomial:

$$
a = 1 \\
b = -6 \\
c = 12 \\
d = -12
$$

Therefore, the polynomial with zeros $2$, $3$, and $-2$ is $f(x) = x^3 – 6x^2 + 12x – 12$.

In this blog post, we have discussed how to form a polynomial with given zeros and degree. We first introduced the concept of a polynomial and its zeros. Then, we discussed the different methods of forming a polynomial with given zeros and degree. Finally, we provided some examples to illustrate the methods.

We hope that this blog post has been helpful in understanding how to form a polynomial with given zeros and degree. If you have any questions or comments, please feel free to contact us.