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Zeros of Polynomial
Zeros of polynomial are the points where the polynomial equals zero on the whole. In simple words, we can say that zeros of polynomial are values of the variable such that the polynomial equals 0 at that point. Zeros of a polynomial are also referred to as the roots of the equation and are often designated as α, β, γ respectively. Some of the methods used to find the zeros of polynomial are grouping, factorization, and using algebraic expressions.
Further, the zeros of polynomial are helpful to form the original polynomial equation. Here we shall learn about how to find the zeros of a polynomial, the sum, and the product of zeros of the polynomial. We will solve a few examples related to it for a better understanding of the concept.
What are Zeros of Polynomial?
The zeros of a polynomial f(x) are the values of x which satisfy the equation f(x) = 0. Here f(x) is a function of x, and the zeros of the polynomial are the values of x for which the f(x) value is equal to zero. The number of zeros of a polynomial depends on the degree of the equation f(x) = 0. All such domain values of the function, for which the range is equal to zero, are called the zeros of the polynomial.
Graphically the zeros of the polynomial are the points where the graph of y = f(x) cuts the xaxis. We shall learn more on this in the below content of representing zeros of a polynomial on the graph.
How to Find Zero of a Polynomial?
There are numerous methods to find the zeros of a polynomial. The number of zeros of the polynomial depends on the degree of the polynomial equation. The different equations have been classified as linear equations, quadratic equation, cubic equation, and higher degree polynomials and each of the equations are individually analyzed to find the zeros of the polynomial. The different types of equations and the methods to find their zeros of polynomial are as follows.
Linear Equation: A linear equation is of the form y = ax + b. The zero of this equation can be calculated by substituting y = 0, and on simplification we have ax + b = 0, or x = b/a.
Quadratic Equation: There are two methods to factorize a quadratic equation. The quadratic equation of the form x^{2} + x(a + b) + ab = 0 can be factorized as (x + a)(x + b) = 0, and we have x = a, and x = b as the zeros of the polynomial. And for a quadratic equation of the form ax^{2}+ bx + c = 0, which cannot be factorized, the zeros can be calculated using the formula method, and the formula is x = [ b ± √(b^{2}  2ac) ] / 2a.
Cubic Equation: The cubic equation of the form y = ax^{3} + bx^{2} + cx + d, can be factorized by applying the remainder theorem. As per the remainder theorem, we can substitute any smaller values for the variable x = α, and if the value of y results to zero, y = 0, then the (x  α) is one root of the equation. Further, we can divide the cubic equation with (x  α) using the long division to obtain a quadratic equation. Finally, the quadratic equation can be solved either through factorization or by the formula method to obtain the required two roots of the equation.
Higher Degree Polynomial: The higher degree polynomial equation is of the form y = ax^{n}+ bx^{n  1}+cx^{n  2} + ..... px + q. These higher degree polynomials can be factorized using the remainder theorem to obtain a quadratic equation. And the quadratic equation can be factorized to obtain the final two required factors.
Zeros of Polynomial Formula
As discussed in the previous section, we can find the zeros of different types of polynomials using different ways. For higher degree polynomials, we use the remainder theorem and ultimately come down to a quadratic polynomial for which we use the quadratic formula to find the zeros. So, the formula that we use to find the zeros of a quadratic polynomial ax^{2}+ bx + c = 0 is:
x = [ b ± √(b^{2}  2ac) ] / 2a
Sum and Product of Zeros of Polynomial
The zeros of a polynomial can be easily calculated with the help of:
Sum and Product of Zeros of Polynomial for Quadratic Equation
The sum and product of zeros of a polynomial can be directly calculated from the variables of the quadratic equation, and without finding the zeros of the polynomial. The zeros of the quadratic equation are represented by the symbols α, and β. For a quadratic equation of the form ax^{2} + bx + c = 0 with the coefficient a, b, constant term c, the sum and product of zeros of the polynomial are as follows.
Sum of Zeros of Polynomial = α + β = b/a =  coefficient of x/coefficient of x^{2}
Product of Zeros of Polynomial = αβ = c/a = constant term/coefficient of x^{2}
Sum and Product of Zeros of Polynomial for Cubic Equation
A cubic polynomial is of the form ax^{3} + bx^{2} + cx + d = 0 , has a, b, c as the coefficients, d is the constant term, and α, β, γ are the roots of the cubic polynomial equation.
α + β + γ = b/a =  coefficient of x^{2}/coefficient of x^{3}
αβ + βγ + γα = c/a = coefficient of x/coefficient of x^{3}
αβγ = d/a = constant/coefficient of x^{3}
Forming an Equation from the Zeros of Polynomial
The zeros of polynomial are useful to form the polynomial equation. For the given 'n' number of zeros of a polynomial, the polynomial equation of 'n' degree can be formed. There are two simple steps to form the equation from the zeros of the polynomial. First, find the factors from the zeros of the polynomial. If x = a , then (x  a) is the required factor. Secondly, find the product of these factors to find the required equation. Let us find the equation for a cubic and quadratic equation.
Cubic Equation: Let us take the roots of the polynomial equation as α, β, γ. The factors of the equation are (x  α), (x  β), (x  γ), and the required equation is (x  α)(x  β)(x  γ) = 0.
Quadratic Equation: For a quadratic equation having the two zeros of the equation as α, β, the factors are (x  α), and (x  β). And the required quadratic equation is x^{2}  x(α+ β) + α.β = 0.
Also, we can find the equation of higher degree polynomial, by forming the required factors, and by taking a product of the factors to form the required equation.
Representing Zeros of Polynomial on Graph
A polynomial expression of the form y = f(x) can be represented on a graph across the coordinate axis. The x value is represented on the xaxis and the f(x) or the y value is represented on the yaxis. The polynomial expression can be a linear expression, quadratic expression, or cubic expression, which is based on the degree of the polynomials. A linear expression represents a line, a quadratic equation represents a curve, and a higher degree polynomial represents a curve with uneven bends.
Zeros of a polynomial can be found from the graph by observing the points where the graph line cuts the xaxis. The xcoordinates of the points where the graph cuts the xaxis are the zeros of the polynomial.
Important Notes on Zeros of Polynomial
 The zeros of polynomial are the values of the variable for which the polynomial is equal to 0.
 We can find the zeros of polynomial by determining the xintercepts.
 To find zeros of a quadratic polynomial, we use the quadratic formula.
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Zeros of Polynomial Examples

Example 1: Sam knows that the zeros of a quadratic polynomial are 3 and 5. How can we help to find the equation of the polynomial?
Solution:
The zeros of the quadratic polynomial are 3 and 5.
Let α = 3, and β = 5
Then, we have the sum of the roots = α + β = 2
Product of the roots = α.β = 15
The required quadratic equation is x^{2}  (α + β)x + α.β = 0
x^{2}  2(x) + (15) = 0
x^{2}  2x  15 = 0
Answer: Therefore the equation of the quadratic polynomial is x2  2x  15 = 0

Example 2: Consider the following polynomial: 3x^{3}  2x^{2} + 5x + 1. What is the sum of the squares of the zeroes of a polynomial?
Solution:
The given polynomial expression is 3x^{3}  2x^{2} + 5x + 1
The formulas for the zeros of the cubic polynomials is as follows.
α + β + γ =  coefficient of x^{2}/coefficient of x^{3} = (2)/3 = 2/3
αβ + βγ + γα = coefficient of x/coefficient of x^{3} = 5/3
(α + β + γ)^{2} = α^{2} + β^{2} + γ^{2} + 2(αβ + βγ + γα)
(2/3)^{2} = α^{2} + β^{2} + γ^{2} + 2(5/3)
4/9 = α^{2} + β^{2} + γ^{2} + 10/3
α^{2} + β^{2} + γ^{2} = 4/9  10/3
α^{2} + β^{2} + γ^{2} = 4/9  30/9
α^{2} + β^{2} + γ^{2} = 26/9
Answer: Hence the sum of the squares of the zeros of the polynomial is 26/9

Example 3: What are the zeros of the polynomial function f(x) = x^{3} 12x^{2}+ 20x?
Solution:
The given function is
f(x) = x^{3}  12x^{2} + 20x
Let us take out x as common
f(x) = x(x^{2}  12x + 20)
Now by splitting the middle term
f(x) = x(x^{2} 2x  10x + 20)
So we get
f(x) = x [x(x  2)  10(x  2)]
f(x) = x(x  2)(x  10)
Here
x = 0
x  2 = 0 where x = 2
x  10 = 0 where x = 10
Answer: Therefore, the zeros of polynomial function is x = 0 or x = 10 or x = 2.
FAQs on Zeros of Polynomial
What Is Meant by Zeros of Polynomial?
The zeros of polynomial refer to the values of the variables present in the polynomial equation for which the polynomial equals 0. The number of values or zeros of a polynomial is equal to the degree of the polynomial expression. For a polynomial expression of the form ax^{n} + bx^{n  1} + cx^{n  2} +.... px + q , there are up to n zeros of the polynomial. The zeros of a polynomial are also called the roots of the equation.
How to Find Zeros of Polynomial?
There are a number of methods to find the zeros of the polynomial. The method used to find the zeros of the polynomial depends on the degree of the equation. The polynomial expression is solved through factorization, grouping, algebraic identities, and the factors are obtained. The factors are individually solved to find the zeros of the polynomial. A quadratic equation of the form x^{2} + x(a + b) + ab = 0 has factors (x + a)(x + b) = 0 and the zeros of the quadratic equation are a, b.
How to Find Zeros of Polynomial Graphically?
The zeros of a polynomial can be easily found graphically by locating the points where the graph of the polynomial expression cuts the xaxis. For all the points where the equation line cuts the xaxis, the x coordinate of the point represents the zeros of the polynomial.
How to Find Complex Zeros of Polynomial Function?
The complex zeros of polynomials can be calculated using the complex number formula of i^{2} = 1. The negative roots can also be simplified using the value of i, from complex numbers. For an equation of the form (x + 3)^{2} = 25, finding the square root of the negative number is not possible. Here we use i^{2} = 1, to write (x + 3)^{2} = 25i^{2}, and on simplification we have (x + 3) = + 5i, and the zeros of the polynomial are 3 + 5i, and 3 5i.
What is the Sum of Zeros of Polynomial?
The sum of the zeros of polynomial for a quadratic equation of the form ax^{2} + bx + c = 0, having α, β as its roots, is α + β = b/a = coefficient of x/coefficient of x^{2}. And the sum of zeros of polynomial for a cubic equation of ax^{3} + bx^{2} + cx + d = 0 having the roots α, β, γ is α + β + γ = b/a = coefficient of x^{2}/coefficient of x^{3}
What is the Product of Zeros of Polynomial?
The product of zeros of polynomial for a quadratic equation of the form ax^{2} + bx + c = 0, having α, β as its roots, is αβ = c/a = constant term/coefficient of x^{2}. And the product of zeros of polynomial for a cubic equation of ax^{3} + bx^{2} + cx + d = 0 having the roots α, β, γ is αβγ = d/a = constant term/coefficient of x^{3}
How Many Zeros of Polynomial does y = f(x) have?
The number of zeros of a polynomial depends on the degree of the polynomial expression y = f(x). For a linear equation in one variable, we have only one root. For a quadratic and cubic polynomial, we have two and three zeros of a polynomial respectively.
What Is the Number of Zeros of Polynomial does a Linear Polynomial have?
A linear polynomial has only one zero of polynomial. A linear expression of the form ax + b = 0 has only one value of x = b/a, which is the zero of this linear polynomial
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