Zeros of a Cubic Polynomial: Sum, Product & Examples

Zeros of a cubic polynomial can be defined as the point at which the polynomial becomes zero. A cubic polynomial is a polynomial with the highest power of the variable or degree is 3. The general form of a cubic polynomial is

ax³ + bx² + cx + d = 0,

where a ≠ 0, and a, b, c are the coefficients of x³, x², x and d is the constant term.

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What is Zeros of a Cubic Polynomial?

Zeros of a cubic polynomial is the point at which the polynomial becomes zero. A cubic polynomial can have three zeros because its highest power (or degree) is three. A quadratic polynomial may have no real solution but a cubic polynomial always has at least one real solution. If a cubic polynomial does have three zeros, two or even all three of them may be repeated.

Let us say α, β, and γ be the zeros of the cubic polynomial ax³ + bx² + cx + d = 0, where a ≠ 0 and a, b, c, are the coefficients of x³, x², x and d is the constant term.

Sum of zeros of a cubic polynomial formula: α + β + γ = –b/a

Sum of product of zeros of a cubic polynomial formula: αβ + βγ + γα = c/a

Product of zeros of a cubic polynomial formula: αβγ = –d/a

Zeros of a cubic polynomial will have 4 possibilities:

1. All three zeroes are real and different
   → This means all the answers (zeroes) are real numbers and none of them are the same.
    
2. All three zeroes are real, but two are the same
   → This means two of the answers are the same, and the third one is different. All are real numbers.
    
3. All three zeroes are real and the same
   → This means all three answers are the same real number.
    
4. One zero is real, and the other two are complex (not real)
   → This means only one answer is a real number, and the other two are complex numbers (they include 'i', the imaginary number).

How to find Zeros of a Cubic Polynomial?

Consider a cubic polynomial of the form ax³ + bx² + cx + d = 0, where a. A cubic polynomial can have three zeros because its highest power (or degree) is three.

We can easily find zeros of a cubic polynomial by following the below steps:

1. Step 1: Let (p – q), p, (p + q) be the three zeros of a given cubic polynomial.
    
2. Step 2: Find p using the sum of zeros of a cubic polynomial formula.
    
3. Step 3: Find out the other two zeros by factoring the equation into a quadratic polynomial.

Example: Find the zeros of the cubic polynomial
x³ – 12x² + 39x – 28 = 0

Let (p – q), p, (p + q) be the zeros of a given cubic polynomial.

Then, Sum of zeros of a cubic polynomial = –b/a

⇒ (p + q) + p + (p – q) = –12/1
⇒ 3p = 12
⇒ p = 4.

Now, find out the other two zeros by factorizing the equation into a quadratic polynomial.

x³ – 12x² + 39x – 28 = (x – 4)(x² – 8x + 7)
x³ – 12x² + 39x – 28 = (x – 4)(x – 1)(x – 7)

⇒ x = 1, x = 4, and x = 7 are the three zeros of the given cubic polynomial.

Product of Zeros of a Cubic Polynomial

Let , , and be the zeros of the cubic polynomial , where , and , , are the coefficients of , , and is the constant term. Then, the product of zeros of a cubic polynomial is given as:

Product of zeros of a cubic polynomial = -(constant term)/(coefficient of )

.

Example: Find the product of zeros of the cubic polynomial .

Given polynomial is , then we have

, and .

Product of zeros = -\frac{d}{a}[/latex])

Therefore, the product of zeros of a cubic polynomial is .

Sum of Zeros of a Cubic Polynomial

Let , , and be the zeros of the cubic polynomial , where , and , , are the coefficients of , , and is the constant term. Then, the sum of zeros of a cubic polynomial is given as:

Sum of zeros of a cubic polynomial = -(coefficiant of )/(coefficiant of )

.

Example: Find the product of zeros of the cubic polynomial .

Given polynomial is , then we have

, and .

Sum of zeros =

Therefore, the sum of zeros of the given cubic polynomial is .

Nature of Zeros of a Cubic Polynomial

Consider a cubic polynomial of the form , where . The nature of these zeros can be defined by the use of the discriminant of a cubic polynomial. It is given by the following relation:

Therefore,

(i). When , the cubic polynomial has real zeros and at least one repeated zeros.

(ii). When , the cubic polynomial has three real and distinct zeros.

(iii). When , the cubic polynomial has a pair of complex conjugates and one real zeros.

Example: find the nature of zeros of the cubic polynomial using the discriminant formula.

Given cubic polynomial is , we have

, , , and .

We know,

Since , then the given cubic polynomial has a pair of complex conjugates and one real zero.

Relation between Zeros and Coefficients of a Cubic Polynomial

Zeros of the polynomial are defined as the values of the variable for which the value of the polynomial is zero. Let , , and be the zeros of the cubic polynomial , where , and , , , are the coefficients of , , and is the constant term.

Then the relation between zeros and coefficient of a cubic polynomial is:

1. Sum of zeros =

.

2. Sum of product of zeros =

.

3. Product of zeros =

.

Example: Verify the relationship between zeros and coefficients of if the zeroes are given as , and .

Comparing the given cubic polynomial with

, , , and

Given the zeros: , and

, , and

(i) Verify the sum of the zeros of the cubic polynomial, i.e.

.

(ii) Verify the sum of the product of the zeros of the cubic polynomial, i.e.

.

(iii) Verify the product of the zeros of the cubic polynomial, i.e.

.

Solved Examples of Zeros of a Cubic Polynomial

Example 1: Find the cubic polynomial whose zeros are , and .

Solution: Given that the zeros of a cubic polynomial are , and that means , , and .

We know that the zeroes of a cubic polynomial are denoted by , , and .

(i). Sum of zeros = [coefficient of /coefficient of ]

(ii). Product of zeros = [constant term/coefficient of ]

(iii). Sum of the product of zeros = [coefficient of /coefficient of ]

On comparing the above solutions, we get

, , , and .

Hence, the cubic polynomial is .

Example 2: Find all the zeros of if is a factor.

Solution: Since is a factor of a given cubic equation.

will completely divide the given equation.

Now, =

=

Zeros of are

Hence, the zeros of are , , and .

If you are checking Zeros of a Cubic Polynomial article, also check related maths articles:
Factoring Polynomials	Polynomial Identities Algebra
Long Division of Polynomials	Cubic Polynomials
Polynomial equations	Zero Polynomial

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