Zeros and Coefficient of a Polynomial: An Introduction
A polynomial is a mathematical term for an algebraic expression containing numerous terms. We are aware that a polynomial can be divided into various categories based on its degree, including a linear polynomial $(x)$, a quadratic polynomial $(x^2+6x-8)$, a cubic polynomial $(x^3+4x^2-9x+13)$, and others.
A polynomial's zeros can be easily determined using the degree of the polynomial, as a result of the formula's number of zeros being equal to the degree of the polynomial. Let's talk about how a polynomial's zeros and coefficients relate to one another in this article using more instances that have been solved.
Definition of Zeroes of a Polynomial
When a polynomial is set to zero, its zeroes are the answers to the specified polynomial equation. Depending on the variable's maximum power within the particular polynomial, polynomials are categorised. In Mathematics, if $p(x)$ is a polynomial with variable $x$ and $k$ is any real number, then $p(x)$ is said to have zero at $x = k$.
Coefficients of a Polynomial
A coefficient is an amount or number that is added to a variable; it is often an integer that is multiplied by the variable in front of it. It is assumed that the coefficient of the variables that do not include any integers is $1$. A coefficient can be a real number, an imaginary number, a decimal, a fraction, or both. It can also be positive or negative.
Relationship between Zeroes and Coefficients of a Polynomial
According to the kind of polynomial, explicit formulae can be used to define the relationship between polynomial zeroes and coefficients.
Linear Polynomial: An expression of the form $ax + b$ of degree $1$ is referred to as a linear polynomial. Here, "a" and "b" are constants, and "x" is a variable. The polynomial zero is equal to -b/a, which is equal to the negative of the constant term divided by the coefficient of x.
Quadratic Polynomial: The maximum degree $2$ expressions of the form $ax^2 + bx + c$ is a quadratic polynomial. Here, a is not equal to $0$. $a, b$, and $c$ are constants, while "x" is a variable. If $\alpha$ and $\beta$ are the polynomial's two zeros, then
The sum of zeroes, $\alpha+\beta$ is $\dfrac{-b}{a}$, which is negative of the coefficient of $x$ divided by the coefficient of $x^2$.
The product of zeroes, $\alpha \beta$ is $\dfrac{c}{a}$ which is a constant term divided by the coefficient of $x^2$.
Cubic Polynomial: The cubic polynomial is an expression with the greatest degree $3$ with the form $ax^3 + bx^2+ cx + d$. In this case, the variables "x" and "a," "b," and "c" are constants, and a is not equal to $0$. In the event where the polynomial's three zeros are $\alpha, \beta$, and $\gamma$. Then,
The sum of zeroes, $\alpha+\beta+\gamma$ is $\dfrac{-b}{a}$ which is equal to the negative coefficient of $x^2$ divided by the coefficient of $x^3$.
The sum of the product of zeroes, $\alpha \beta+\beta \gamma+\alpha \gamma$ is $\dfrac{c}{a}$ which is equal to the coefficient of $x$ divided by the coefficient of $x^3$.
The product of zeroes, $\alpha \beta\gamma$ is $\dfrac{-d}{a}$ which is equal to the negative of the constant term divided by the coefficient of $x^3$.
Zeros of Polynomial Formula
In order to get the zeros of a quadratic polynomial $ax^2+ bx + c = 0$, we can apply the following formula:
$x = \dfrac{- b \pm \sqrt{b^{2} - 4ac}}{2a}$
Zeros of a Polynomial Solved Examples
Example 1: Find the zeros of the quadratic polynomial $6{x}^{2} – 13x + 6$.
Solution: We are given $6{x}^{2} – 13x + 6$= $6{x}^{2} – 4x -9x + 6$
=$2x(3x-2)-3(3x-2)$
=$(2x-3)(3x-2)$
Put the value of the given expression equal to zero
Therefore, $(2x-3)=0$ or $(3x-2)=0$
Hence the zeros of the given polynomial are
$x=\dfrac{3}{2}$ or $x=\dfrac{2}{3}$
Example 2: What will be the answer if the quadratic polynomial $4x^2 - 9$ should have its zeros' sum and product evaluated?
Solution: The quadratic polynomial is $4x^2 – 9$.
We can write $4x^2 – 9$ as $2x^2 – 3^3$, which is equal to $(2x+3)(2x-3)$.
To find the zeros of a polynomial, equate the above expression to $0$
$(2x+3)(2x-3) = 0$
$2x+3 = 0$
$2x = -3$
$x = \dfrac{-3}{2}$
Similarly, $2x-3 = 0$,
$2x = 3$
$x = \dfrac{3}{2}$
Therefore, the two zeros of the given quadratic polynomial come out to be $\dfrac{3}{2}$ and $\dfrac{-3}{2}$.
Sum and product of a polynomial:
The sum of the zeros $= \dfrac{3}{2}+ \dfrac{-3}{2} =\dfrac{3}{2}-\dfrac{3}{2} = 0$
The product of zeros $= \dfrac{3}{2}.\dfrac{-3}{2} = \dfrac{-9}{4}$.
Practice Questions
Question 1: In the quadratic polynomial 9x2 – 16 + 20, find the sum and product of the zeros.
Question 2: Determine the polynomial 5x - 10's zero.
Answer:
The sum and product of zeros of the given polynomial came out to be 16/9 and 20/9.
The zero of the given polynomial came out to be 2.
Summary
As discussed above, an algebraic expression containing multiple terms is called a polynomial. Polynomials come in a variety of shapes, including linear, quadratic, cubic, and others. This article provides you with a simple definition of zeros of polynomials. The relationship between the zeroes and the coefficients is discussed thoroughly. Some solved examples are provided in this article to boost your concept and clear your doubts about the zeros of polynomials.
FAQs on Relationship between Zeros and Coefficient of a Polynomial
1. What does the degree of a polynomial function mean?
The largest power of a variable in a polynomial equation is the polynomial's degree. Only terms with variables are taken into consideration when calculating the degree of a polynomial function. The degree of a polynomial is determined by the variable term's highest exponential power. Any polynomial's degree is determined solely by its variables; coefficients should not be taken into account. For example, $4x^3+6x$ has degree 3.
2. What is polynomial factoring?
Splitting the given equation into its factor expressions and then writing each of those expressions as a product is how polynomials are factored. To factor $x^2 + 2x$, for instance, we divide it into two factors, x and (x + 2), then write it as the product of these two parts: $x^2 + 2x$ = $x(x + 2)$. Here, factoring contains higher-degree polynomials as well as the ideas of the largest common factor, factor theorem, and long division.
3. What is the relationship between coefficient and roots?
The link between coefficients and roots is, The negative of the coefficient of the second term divided by the coefficient of the first term is equal to the sum of the roots in a quadratic equation.
While the root product is represented by the third term divided by the first term. Let us assume a polynomial $a{x}^{2} + bx + c$ having $\alpha$ and $\beta$ be the zeros of the polynomial then, the sum of the zeros is given as $(\alpha+\beta) = \dfrac{- b + \sqrt{b^{2} - 2ac}}{2a}+\dfrac{- b - \sqrt{b^{2} - 2ac}}{2a} = \dfrac{- b}{a}$.
The product of zeros is given as $(\alpha \beta) = \dfrac{- b + \sqrt{b^{2} - 2ac}}{2a} \times \dfrac{- b - \sqrt{b^{2} - 2ac}}{2a} =\dfrac{ c}{a}$.
