How to Find Zeros from the Coefficients: Step-by-Step Methods
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 Understand the Relationship Between Zeroes and Coefficients of Polynomials
1. What is the basic relationship between the zeros and coefficients of a quadratic polynomial?
For a standard quadratic polynomial, expressed as ax² + bx + c, where a ≠ 0, there are two key relationships with its zeros (let's call them α and β):
The Sum of the Zeros (α + β) is equal to -b/a, which is the negative of the coefficient of x divided by the coefficient of x².
The Product of the Zeros (αβ) is equal to c/a, which is the constant term divided by the coefficient of x².
2. How does the relationship between zeros and coefficients work for a cubic polynomial?
For a cubic polynomial of the form ax³ + bx² + cx + d, with zeros α, β, and γ, the relationship expands to three formulas:
Sum of the zeros: α + β + γ = -b/a
Sum of the products of zeros taken two at a time: αβ + βγ + γα = c/a
Product of the zeros: αβγ = -d/a
3. How do you create a quadratic polynomial if the sum and product of its zeros are provided?
You can construct a quadratic polynomial using a direct formula if you know the sum (S) and product (P) of its zeros. The general form of the polynomial is k[x² - (S)x + P], where 'k' is any non-zero real number. For example, if the sum of zeros is 4 and the product is 3, the polynomial would be k(x² - 4x + 3).
4. What is the fundamental logic that explains why this relationship between zeros and coefficients exists?
The relationship exists because of the Factor Theorem. If α and β are the zeros of a quadratic polynomial, then (x - α) and (x - β) must be its factors. The polynomial can be written as p(x) = k(x - α)(x - β). When you expand this, you get p(x) = k[x² - (α + β)x + αβ]. By comparing the coefficients of this expanded form with the standard form ax² + bx + c, we can directly equate the terms and derive the relationships for the sum and product of zeros.
5. What is the connection between the degree of a polynomial and the number of zeros it can have?
The connection is defined by the Fundamental Theorem of Algebra. The theorem states that a polynomial of degree 'n' will have exactly 'n' zeros. These zeros can be real or complex numbers, and some may be repeated. For the scope of Class 10 Maths, this means a polynomial of degree 'n' can intersect the x-axis at a maximum of n points. For instance, a quadratic polynomial (degree 2) can have at most two real zeros, and a cubic polynomial (degree 3) can have at most three.
6. How are the zeros of a polynomial represented on its graph?
The real zeros of a polynomial are the x-coordinates of the points where its graph intersects or touches the x-axis. For a quadratic polynomial like ax² + bx + c, the shape of the graph is a parabola. The points where this parabola crosses the x-axis represent the zeros (α and β). The coefficients 'a', 'b', and 'c' determine the shape and position of this parabola, thereby dictating the location of these x-intercepts.
7. If you change only the constant term 'c' in a quadratic polynomial, how does it affect the sum and product of its zeros?
Changing only the constant term 'c' in ax² + bx + c has a specific impact:
The Sum of Zeros (α + β = -b/a) will remain unchanged because it does not depend on 'c'.
The Product of Zeros (αβ = c/a) will change directly in proportion to the change in 'c'.
Graphically, this means the parabola (the graph of the polynomial) shifts vertically up or down without changing its axis of symmetry.
