Formula Proof and Solved Examples of Zeroes and Coefficients of Polynomials
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 Zeroes and Coefficients of Polynomials Explained
1. What is the relationship between zeroes and coefficients of a polynomial?
The relationship between zeroes and coefficients of a polynomial gives formulas that connect the roots of a polynomial to its coefficients. For a quadratic polynomial ax² + bx + c with zeroes α and β:
- α + β = −b/a
- αβ = c/a
2. What is the formula for the sum and product of zeroes of a quadratic polynomial?
The sum and product of zeroes of a quadratic polynomial ax² + bx + c are given by simple coefficient formulas. If α and β are the zeroes:
- Sum of zeroes (α + β) = −b/a
- Product of zeroes (αβ) = c/a
3. How do you find zeroes of a quadratic polynomial using coefficients?
You can find the zeroes of a quadratic polynomial using the quadratic formula based on its coefficients. For ax² + bx + c = 0, the zeroes are:
- x = [−b ± √(b² − 4ac)] / 2a
- Identify values of a, b, and c.
- Substitute into the quadratic formula.
- Simplify to get the two zeroes.
4. Can you give an example of the relationship between zeroes and coefficients?
Yes, the relationship can be shown clearly using a simple quadratic example. Consider the polynomial 2x² − 7x + 3.
- Here, a = 2, b = −7, c = 3.
- Sum of zeroes = −b/a = 7/2
- Product of zeroes = c/a = 3/2
5. What is the relationship between zeroes and coefficients of a cubic polynomial?
For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β, and γ, the relationships are based on symmetric sums. They are:
- α + β + γ = −b/a
- αβ + βγ + γα = c/a
- αβγ = −d/a
6. Why is the sum of zeroes equal to −b/a in a quadratic polynomial?
The sum of zeroes equals −b/a because of how a quadratic polynomial is formed from its roots. If the zeroes are α and β, then:
- Polynomial = a(x − α)(x − β)
- = a[x² − (α + β)x + αβ]
- α + β = −b/a
7. How do you form a quadratic polynomial if the zeroes are given?
A quadratic polynomial with zeroes α and β is formed using the formula x² − (α + β)x + αβ. Steps:
- Find the sum (α + β).
- Find the product (αβ).
- Substitute into x² − (sum)x + product.
- Sum = 5, Product = 6
- Required polynomial = x² − 5x + 6
8. What is the relationship between roots and coefficients in general polynomial theory?
In general polynomial theory, the relationship between roots and coefficients is given by Vieta’s formulas, which express sums and products of roots in terms of coefficients. For a polynomial:
- aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀
9. How can you verify the relationship between zeroes and coefficients?
You can verify the relationship by calculating zeroes and checking their sum and product against coefficient formulas. Steps:
- Find the zeroes using factorization or quadratic formula.
- Compute their sum and product.
- Compare with −b/a and c/a.
- Sum = 5 = −(−5)/1
- Product = 6 = 6/1
10. What are common mistakes when applying the relationship between zeroes and coefficients?
Common mistakes usually involve incorrect signs or ignoring the leading coefficient a. Key points to remember:
- Always divide by a in formulas like −b/a and c/a.
- Be careful with negative signs.
- Ensure the polynomial is written in standard form ax² + bx + c.
