Cubic Polynomial

A cubic polynomial is a type of polynomial in which the highest power of the variable, or degree, is 3. It typically has up to four terms. 

Here are a few examples of cubic polynomials:

• 2x³ + 5x² + 3x + 4
• -x³ + 6x² + x – 1
• x³ + 1

Formula

The general form of a cubic polynomial is: 

ax³ + bx² + cx + d 

Here, 

• x is a variable 
• a, b, c, and d are real numbers
• a ≠ 0

Thus, the cubic polynomial equation takes the form ax³ + bx² + cx + d = 0. The values of x that satisfy the cubic equation are called the roots or zeros of the cubic polynomial. 

Finding Zeros or Roots

Let us find the zeros of the cubic equation x³ – 6x² + 11x = 6

Arranging in the Standard Form 

x³ – 6x² + 11x = 6

⇒ x³ – 6x² + 11x – 6 = 0

Breaking Down to the Product of Linear and Quadratic Factors

⇒ x³ – x² – 5x² + 5x + 6x – 6 = 0

⇒ x²(x – 1) – 5x(x – 1) + 6(x – 1) = 0

⇒ (x – 1)(x² – 5x + 6) = 0 …..(i)

Solving the Quadratic Factor

⇒ (x – 1)(x² – 3x – 2x + 6) = 0

⇒ (x – 1)[x(x – 3) – 2(x – 3)] = 0

⇒ (x – 1)(x – 3)(x – 2) = 0 …..(ii)

Solving for x

Now, from (ii), we get

(x – 1) = 0 ⇒ x = 1

(x – 2) = 0 ⇒ x = 2

(x – 3) = 0 ⇒ x = 3

Thus, the roots or zeros are x = 1, 2, and 3

A cubic polynomial of the form of P(x) = ax³ + bx² + cx + d (a ≠ 0) typically has 3 roots. 

Alternate Method: Using Formulas

If the roots of a cubic polynomial are p, q, and r, they satisfy these relations: 

• ${p+q+r=-\dfrac{b}{a}}$
• ${pq+qr+rp=\dfrac{c}{a}}$
• ${pqr=-\dfrac{d}{a}}$

Let us find the roots of x³ – 6x² + 11x = 6 using formulas, assuming the roots are in arithmetic progression.

Since the roots are in arithmetic progression, the roots p, q, and r are x – d, x, and x + d, respectively.

Finding the Linear Factor

From the equation, we get

x³ – 6x² + 11x = 6

⇒ x³ – 6x² + 11x – 6 = 0 …..(i)

Now, comparing equation (i) with the standard form of the cubic equation, we get

a = 1, b = -6, c = 11, and d = -6

Using the cubic formula for the sum of roots, we get

${p+q+r=-\dfrac{b}{a}}$

⇒ ${x-d+x+x+d=-\dfrac{\left( -6\right) }{1}}$ 

⇒ 3x = 6

⇒ x = ${\dfrac{6}{3}}$

⇒ x = 2 …..(ii)

Thus, x = 2 is one of the roots. We can find the other roots by factoring the cubic equation into a quadratic equation.

Finding Other Factors by Synthetic Division

By factoring a cubic polynomial, we get simpler polynomials whose product gives the original function. Each simpler polynomial is a factor of the cubic polynomial.

According to the factor theorem, (x – a) is a factor of the polynomial P(x) of degree n ≥ 1 if and only if P(a) = 0, where a is any real number. 

The formula of the factor theorem is given by P(x) = (x – a) Q(x)

If P(x) is a cubic polynomial, then Q(x) is a quadratic polynomial. Now, using synthetic division, we can divide P(x) by (x – a) to find the factors of P(x).

Now, when factoring x³ – 6x² + 11x – 6 = 0 with (x – 2) as one of the factors, we can find the other factor Q(x) by synthetic division.

Here, the quotient Q(x) is x² – 4x + 3 (one order less than the given polynomial), and the remainder is 0

Thus, x³ – 6x² + 11x – 6 is factored as (x – 2)(x² – 4x + 3) …..(iii)

Factoring the Quadratic Quotient

Now, Q(x) can further be factored by the quadratic formula.

x = ${\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$

⇒ x = ${\dfrac{-\left( -4\right) \pm \sqrt{\left( -4\right) ^{2}-4\times 1\times 3}}{2\times 1}}$

⇒ x = ${\dfrac{4\pm \sqrt{16-12}}{2}}$

⇒ x = ${\dfrac{4\pm \sqrt{4}}{2}}$

⇒ x = ${\dfrac{4\pm 2}{2}}$

⇒ x = 3 and 1 …..(iv)

Thus, from (ii) and (iv), we get 

x = 1, 2, and 3 are the roots of x³ – 6x² + 11x = 6

The quadratic part can also be factored by the middle-term method.

Graphing

When a cubic polynomial cannot be solved using the methods mentioned above, it can be solved graphically. The points where the graph intersects the x-axis (x-intercepts) represent the solutions, also known as the roots of the cubic polynomial. 

While plotting a graph for a cubic polynomial:

• If the coefficient of x³ (denoted by ‘a’) is positive, the graph will extend from down to up.
• If the coefficient of x³ (denoted by ‘a’) is negative, the graph will extend from up to down.

A general cubic polynomial graph is shown:

Solved Examples