Relation Between Coefficients and Zeros of a Polynomial

Introduction

Polynomials are mathematical expressions containing variables and coefficients. James Waddell Alexander II invented the concept of the polynomial. There are various terms associated with the polynomial. In this tutorial, we will discuss the meaning of polynomial, various correlations between the zeroes and the coefficients of polynomial equations with solved examples.

Polynomials

Polynomials are defined as the algebraic expressions containing one or more variable terms multiplied by constant terms. There are two terms associated with a polynomial, such as coefficients (i.e., constants) and variables. For example, $\mathrm{\mathit{f}(p)=p^2+2p+5}$ is an example of a polynomial. The given polynomial is denoted by f(p). Here, p and p² are the variable terms and they are multiplied with constants. The word “polynomial” is derived from two words, i.e., “poly” (means many) and “nominal” (means term) The polynomials are extensively used in various sectors, including chemistry, physics, mathematics, engineering, and social science.

Zeros of Polynomials

Zeros of polynomials are defined as the numeric values for which the polynomial expression becomes zero. In other words, the roots of the polynomial are known as zeroes of the polynomial. It is necessary to know the zeroes of a polynomial as it determines the degree of the polynomial. There are various types of polynomial expressions depending on their degree. For example, m is said to be the zero of the polynomials. f(p) if f(m)=0.

Coefficients of Polynomials

We have seen that a polynomial expression contains two different types of terms: constant terms and variables. The constant terms are known as the coefficients of the polynomial. The coefficients are generally integer values. However, they also can be fractional, decimal, or imaginary numbers. If no constant is associated or multiplied with the variable, then the coefficient is 1. Let’s consider an example of a polynomial expression, i.e., $\mathrm{\mathit{f}(p)=4p^3-p+0.5}$. In this case the coefficients are 4, -1, and 0.5.

Relation Between Zeros and Coefficient of Linear Polynomials

In mathematics, linear polynomials are the polynomials containing only one variable having the highest degree 1. The mathematical expression for a linear polynomial can be written as

$$\mathrm{\mathit{f}(p)=np+q}$$

where p is the only variable. In addition, n (≠0) and 1 are the coefficients. Since zeros represent the degree of the polynomial; therefore, there exists one zero for any arbitrary linear polynomial. The zeros of the polynomial $\mathrm{=\frac{-q}{n}=\frac{-(constant\: term)}{coefficient\: of\: p}}$

Relation Between Zeros and Coefficient of Quadratic Polynomials

In mathematics, quadratic polynomials are the polynomials containing one or more variables having the highest degree 2. It is also called a quadratic function. The mathematical expression for a quadratic polynomial can be written as

$$\mathrm{\mathit{f}(p)=mp^2+np+q}$$

where p² and p are the variables. In addition, m (≠0), n, and q are the coefficients.

Since zeros represent the degree of the polynomial; therefore, there exists two zeros for any arbitrary quadratic polynomial. Let’s assume α and β are the zeros of the quadratic polynomials. The relationships between the zeros and the coefficients of the quadratic polynomial are summarized below.

• Addition of zeros − $\mathrm{α+β=\frac{-n}{m}=\frac{-(coefficient\: of\: p)}{coefficient\: of\: p^2}}$

• Multiplication of zeros − $\mathrm{α\times β=\frac{q}{m}=\frac{constant\: term}{coefficient\: of\: p^2}}$

Relation Between Zeros and Coefficient of Cubic Polynomials

In algebra, cubic polynomials are the polynomials containing one or more variables having the highest degree 3. It is also called cubic function. The mathematical expression for a cubic polynomial can be written as

$$\mathrm{\mathit{f}(p)=mp^3+np^2+qp+r}$$

where p³, p² and p are the variables. In addition, m (≠0), n, q, and r are the coefficients.

Since zeros represent the degree of the polynomial; therefore, there exists three zeros for any arbitrary cubic polynomial. Let’s assume α, β, and γ are the zeros of the quadratic polynomials. The relationships between the zeros and the coefficients of the cubic polynomial are summarized below.

• Addition of zeros − $\mathrm{α+β+λ=\frac{-n}{m}=\frac{-(coefficient\: of\: p^2)}{coefficient\: of\: p^3 }}$

• Multiplication of zeros − $\mathrm{α\times β\times λ=\frac{-r}{m}=\frac{-(constant\: term)}{coefficient\: of\: p^3 }}$

• Sum of the product of zeros − $\mathrm{ αβ+βλ+αλ=\frac{q}{m}=\frac{coefficient\: of\: p}{coefficient\: of\: p^3 }}$

Solved Examples

Example 1

Evaluate the addition and multiplication of the zeros of the following polynomial.

$$\mathrm{\mathit{f}(p)=0.25p^2-p+1}$$

Solution

The given quadratic polynomial is

$$\mathrm{\mathit{f}(p)=0.25p^2-p+1}$$

Comparing with the standard form of quadratic polynomial, we get

m = 0.25, n = -1, q =1

The addition of zeros = $\mathrm{\frac{-n}{m}=\frac{-(-1)}{0.25}=4}$

The multiplication of zeros =$\mathrm{\frac{q}{m}=\frac{1}{0.25}=4}$

∴ The addition and multiplication of the zeros of the given polynomial expression are 4 and 4, respectively.

Example 2

Evaluate the sum and product of the zeros of the following polynomial. Also, find the sum of the product of zeros.

$$\mathrm{\mathit{f}(p)=(-27p^3)-9p^2+6p}$$

Solution

The given cubic polynomial is

$$\mathrm{\mathit{f}(p)=(-27p^3)-9p^2+6p}$$

Comparing with the standard form of a cubic polynomial, we get

m = -27, n = -9, q =6, r = 0

The addition of zeros =$\mathrm{\frac{-n}{m}=\frac{-(-9)}{27}=1/3}$

The multiplication of zeros $\mathrm{=-\frac{r}{m}=\frac{0}{27}=0}$

The sum of the product of zeros = $\mathrm{\frac{q}{m}=\frac{6}{-27}=\frac{-2}{9}}$

∴ The addition and multiplication of the zeros of the given polynomial expression are $\mathrm{\frac{1}{3}}$ and 0, respectively. Also, the sum of the product of zeros is $\mathrm{\frac{-2}{9}}$.

Example 3

Evaluate the zero of the polynomial −

$$\mathrm{\mathit{f}(p)=0.5p-25}$$

Solution

The given linear polynomial is

$$\mathrm{\mathit{f}(p)=0.5p-25}$$

Comparing with the standard form of a linear polynomial, we get

n = 0.5, q = -25

The zero of the polynomial is =$\mathrm{\frac{-q}{n}=\frac{-(-25)}{0.5}=50}$

∴ The zero of the polynomial is 50.

Word Problems

Problem 1: Evaluate the addition and multiplication of the zeros of the following polynomial.

$$\mathrm{\mathit{f}(p)=1.5p^3+4p^2-7.5p+2}$$

Problem 2: Evaluate the addition and multiplication of the zeros of the following polynomial.

$$\mathrm{\mathit{f}(p)=2p^2-5p-3}$$

Conclusion

The present tutorial gives a brief introduction about the relation between the coefficients and zeros of the polynomial. The basic meaning of the polynomial and associated terms have been briefly described. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of the relation between the coefficients and zeros of the polynomial.

FAQs

1. What will be the product of zeros if one of the roots of a polynomial is zero?

If one of the roots of the polynomial is 0, then the product of zeros will be zero.

2. How many zeros are there in a polynomial?

The number of zeros depends upon the degree of the polynomial.

3. Can we determine the zeros of a polynomial having a degree of more than three?

Yes. We can determine the zeros by finding the roots of the polynomial having a degree of more than three. However, the calculation procedure is lengthy and time-consuming.

4. Can we determine the polynomial expression if the sum and product of zeros are given?

Yes. Let’s consider the sum and product of a quadratic polynomial are given as α+β and αβ, respectively. Then the polynomial expression can be obtained using the following formula

$$\mathrm{\mathit{ f}(p)=p^2-(α+β)p+αβ}$$

$$\mathrm{or \mathit{f}(p)=p^2-(sum\: of\: zeros)p+product\: of\: zeros}$$

5. What are the applications of polynomial expressions?

Polynomials are extensively used in various sectors, including chemistry, physics, mathematics, engineering, and social science.