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 relationship between zeroes and coefficients of a polynomial?
The relationship between zeroes and coefficients of a polynomial is established by Vieta's formulas, which link the sum and product of the zeroes to the polynomial's coefficients. For a quadratic polynomial, these relationships help find unknown coefficients if zeroes are given.
2. How do you find the sum and product of zeroes in a quadratic polynomial?
For a quadratic polynomial $ax^2 + bx + c$, the sum of its zeroes is given by $-\frac{b}{a}$, and the product is $\frac{c}{a}$. These formulas directly relate zeroes to the coefficients 'a,' 'b,' and 'c.'
3. Why are zeroes of a polynomial important in mathematics?
Zeroes of a polynomial are the values of x where the polynomial equals zero. They are important because they represent solutions to equations, help in factorization, graph plotting, and understanding how coefficients influence the graph's intersection with the x-axis.
4. What are Vieta's formulas in the context of polynomials?
Vieta’s formulas relate the sum and product of the roots (zeroes) of a polynomial to its coefficients. For quadratics, these are $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$, where $\alpha$ and $\beta$ are roots.
5. Can you derive a quadratic polynomial if zeroes are known?
If the zeroes of a quadratic polynomial are known, say $p$ and $q$, the polynomial can be written as $a(x - p)(x - q)$. Expanding this form gives you the standard $ax^2 + bx + c$ format, showing the direct connection to coefficients.
6. How are coefficients affected if all zeroes of a polynomial are doubled?
If all zeroes of a polynomial are doubled, the coefficients change in a specific way. The new polynomial is obtained by replacing $x$ with $\frac{x}{2}$, causing each term's coefficient to be adjusted by the necessary power of 2.
7. What is the relationship between zeroes and coefficients for cubic polynomials?
For a cubic polynomial $ax^3 + bx^2 + cx + d$, Vieta’s formulas state:
- Sum of zeroes $= -\frac{b}{a}$
- Sum of product of zeroes taken two at a time $= \frac{c}{a}$
- Product of zeroes $= -\frac{d}{a}$
8. Why do the sum and product of zeroes matter for solving polynomials?
Knowing the sum and product of zeroes helps in reconstructing the polynomial, checking solutions, and solving equations efficiently. These relationships simplify calculations and allow you to move between roots and coefficients easily for polynomial equations.
9. How can you check whether a given number is a zero of a polynomial?
To check if a number is a zero of a polynomial, substitute it for $x$ in the polynomial equation. If the result is zero, then that number is a zero. This method directly tests the polynomial's roots.
10. Can the relationship between zeroes and coefficients be used to factor polynomials?
Yes, the relationship between zeroes and coefficients helps factor polynomials. If zeroes are known, the polynomial can be expressed as the product of linear factors, each corresponding to a root, making factorization straightforward.
11. Does this relationship apply to polynomials of all degrees?
The relationship between zeroes and coefficients extends to any degree. As the degree increases, Vieta’s formulas include more terms, linking all combinations of zeroes to the coefficients, thus ensuring this relationship applies generally.
