Zeros of a Cubic Polynomial

The zeros of a cubic polynomial are the points at which the polynomial becomes zero. Since the highest degree of a cubic polynomial is three, it can have up to three zeros. A cubic polynomial always has at least one real solution, whereas a quadratic polynomial might not have any. Some or all zeros may be repeated.
In the cubic polynomial \(ax^3 + bx^2 + cx + d = 0 \), where a ≠ 0 and a, b, c are the coefficients of x3, x2, x, and d is the constant term, let α, β, and γ be the zeros.

Zeros of Cubic Polynomial Formulas
A cubic polynomial formula’s sum of zeros is \(α+β+γ= -b/a\)
A cubic polynomial formula’s sum of the products of zeros is \(αβ+βγ+γα=c/a\)
A cubic polynomial formula’s product of zeros is \(αβγ= -d/a\)


Properties of Zeros of a Cubic Polynomial
Since its degree is 3, a cubic polynomial can have at most three zeros.
Even if the other two are complex, every cubic polynomial with real coefficients will have at least one real zero.
For a cubic polynomial \(ax^3 + bx^2 + cx + d \) with zeros p, q, r:
\(p + q + r = -\frac{b}{a} \)  it is the sum of zeros.
\(pq + qr + rp = \frac{c}{a} \) it is the sum of the products taken two at a time.
\(pqr = -\frac{d}{a} \) it is the product of the zeros.
 

Consider a cubic polynomial with the formula \(ax^3 + bx^2 + cx + d = 0 \), where a. A cubic polynomial can have three zeros because its highest power (or degree) is three.  Since three is a cubic polynomial’s highest power (or degree), it is possible for it to contain three zeros.


By taking the actions listed below, we can quickly determine a cubic polynomial’s zeros:


Step 1: Assume that the three zeros of a given cubic polynomial are (p - q), p, and (p + q).

Step 2: Utilizing a cubic polynomial formula’s sum of zeros, find p.

Step 3: Factor the equation into a quadratic polynomial to determine the remaining two zeros.

For example, find the zeros of the cubic polynomial
\(x^3 - 12x^2 + 39x - 28 = 0 \)
Let the zeros of a given cubic polynomial be (p - q), p, and (p + q). Then, a cubic polynomial’s sum of zeros equals -b / a.

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

Now factorize the equation into a quadratic polynomial to determine the remaining two zeros.

\(x^3 - 12x^2 + 39x - 28 = (x - 4)(x^2 - 8x + 7) \)
\( x^3 - 12x^2 + 39x - 28 = (x - 4)(x - 1)(x - 7) \)

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

Let α, β, and γ be the cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), where a ≠ 0 and a, b, and c are the coefficients of \(x^3\), \(x^2\), and x, and d is the constant term. Next, a cubic polynomial’s product of zeros is provided as follows:

Product of zeros of a cubic polynomial = \( -\dfrac{\text{constant term}}{\text{coefficient of } x^3} \).

⇒ \( \alpha \beta \gamma = -\frac{d}{a} \).

Example: Determine the cubic polynomial \(\ kx^3 - 5x^2 - 12x + k = 0 \ \)
.
If the polynomial is \(\ kx^3 - 5x^2 - 12x + k = 0 \), then we have a = k, and d = k.

Product of zeros = -(constant term)/(coefficient of \(x^3\))

⇒\(\ \alpha \beta \gamma = -\frac{k}{k} = -1 \ \)

Therefore, the product of the zeros of a cubic polynomial is -1.


Sum of Zeros of a Cubic Polynomial
Let α, β, and γ be the zeros of the cubic polynomial \(\ ax^3 + bx^2 + cx + d = 0 \ \) where a ≠ 0, where a, b, and c are the coefficients of \(x^3\),\(x^2\), and x and d is the constant term. Next, a cubic polynomial’s sum of zeros is expressed as follows:

\(\ \text{The sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \)

⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).

Example: Determine the cubic polynomial \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \).

\(\ \text{Sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \)
⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).

If the polynomial is \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \), then we have
a= 5 and b = -15.

Sum of zero = -ba

⇒ \(\ \alpha + \beta + \gamma = -\left(-\frac{15}{5}\right) = 3 \ \)

Therefore, the given cubic polynomial’s sum of zeros is 3.
 

Consider a cubic polynomial with the formula \(\ ax^3 + bx^2 + cx + d = 0 \ \), where a ≠ 0. The nature of these zeros can be defined by the use of the discriminant of a cubic polynomial. The following relation proves it:
\(\ D = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd \ \)
Therefore,

1. The cubic polynomial contains real zeros and at least one repeated zero when D = 0.
    
2. The cubic polynomial has three distinct and real zeros when D > 0.
    
3. The cubic polynomial has one real zero and two complex conjugates when 
    

D < 0.
For example,
Determine the type of zeros in the cubic polynomial \( x^3+x=0\). Applying the discriminant formula.
Given that \(x^3+x=0\) is a cubic polynomial, we have 

a = 1, b = 0, c = 1, and d = 0.

We know, \(\ D = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd \ \)
So,

⇒\(\ D = (0)^2(1)^2 - 4(1)(1)^3 - 4(0)^3(0) - 27(1)^2(0)^2 + 18(1)(0)(1)(0) \ \)

⇒\(\ D = 0 - 4 - 0 - 0 + 0 \ \)

⇒\(D=-4<0\)

The given cubic polynomial has one real zero and two complex conjugates since 

D < 0.
 

The values of the variable for which the polynomial’s value is zero are known as zeros of the polynomial. Let α, β, and γ be the zeros of the cubic polynomial \(\ ax^3 + bx^2 + cx + d = 0 \ \), where a ≠ 0, and a, b, and c are the coefficients of \(x^3\), \(x^2\), x and d is the constant term.


Then, a cubic polynomial’s zeros and coefficients have the following relationship:


Sum of zeros = -ba
⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).


Sum of the product of zeros = ca.
⇒\(\ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \ \).


Product of zeros = -da
⇒\(\ \alpha\beta\gamma = -\frac{d}{a} \ \).


For Example:
Check the connection between zeros and coefficients of \(\ x^3 + 5x^2 - 6x = 0 \ \) if the zeros are given as -6, -0, and -1.
Comparing the given cubic polynomial with \(\ ax^3 + bx^2 + cx + d = 0 \ \)


⇒a=1, b=5, c=-6, and d=0
Given the zeros: -6, -0, and -1


⇒α=-6,β=0, and γ=1
Check that the cubic polynomial’s zeros sum is correct, i.e.


\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \)
⇒\(\ (-6) + 0 + 1 = -\frac{5}{1} \ \)
⇒ -5= -5.


Confirm the total product of the cubic polynomial’s zeros, i.e.
\(\ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \ \)
⇒\( (-6)(0) + (0)(1) + (1)(-6) = -6 \)
⇒-6=-6


Confirm that the cubic polynomial’s zeros are multiplied, that is,
\( \alpha\beta\gamma = -\frac{d}{a} \)
⇒\((-6)(0)(1)=-01\)
⇒ 0 = 0.
 

Mastering the zeros of a cubic polynomial helps in understanding the relationship between factors and roots. It also strengthens algebraic problem-solving and graph interpretation skills.

• Always start by checking for common factors in the polynomial before proceeding with other methods it simplifies your work.
  
   
• Use the rational root Theorem to test possible rational zeros; it saves time by narrowing down potential values.
  
   
• Once you find one zero, use synthetic or long division to reduce the cubic equation to a quadratic and solve easily.
  
   
• Remember that the sum and product of zeros relate to the coefficients of the polynomial, helping in quick verification.
  
   
• Practice factoring and substitution regularly; familiarity with patterns like \(( 𝑥 − 𝑎 ) ( 𝑥 − 𝑏 ) ( 𝑥 − 𝑐 ) (x−a)(x−b)(x−c)\) improves accuracy and speed.

Zeros of a cubic polynomial are used in areas like engineering, physics, and more. Let us see how the zeros of cubic polynomials help in real life.
 

• Engineering - To simulate the form of suspension cables in bridges, engineers employ cubic polynomials. The zeros of the polynomial show the spots where the cable really touches the ground, such as in a hanging wire or bridge support. In design, identifying the zeros helps in identifying stress and anchor points.
  
   
• Physics - A cubic equation can be used to model the height of a thrown ball over time. The times the ball touches the ground are indicated by the zeros. This helps in determining the flight’s maximum range and duration.
  
   
• Profit models for business - A business uses a cubic function to model its profit over time.
  \(\ P(x) = -2x^{3} + 15x^{2} - 30x + 10 \ \) This function’s zeros indicate break-even points, or times when the profit is zero. When the profit is zero. For financial planning, this is helpful.
  
   
• Speed optimization in automotive design - A cubic equation can be used to model the relationship between vehicle speed and fuel efficiency. The zeros represent the speeds at which fuel efficiency is zero or ineffective. This helps in design improvement for increased mileage.
  
   
• Environmental science - Cubic polynomials are used to simulate the water flow rate in a dam or river over time. When the flow rate is zero, the zeros alert scientists, which may indicate a flood or drought warning.