Cubic Polynomial

A cubic polynomial is a type of polynomial with the highest power of the variable or degree to be $3$. Cubic polynomials are used in various areas of mathematics and science, including physics, engineering, and economics.

Definition

A cubic polynomial is a polynomial with the highest exponent of a variable i.e. degree of a variable as $3$. The general form of a cubic polynomial is $ax^{3}+bx^{2}+cx+d=0$, $a\neq 0$, where $a,b,c$ are coefficients and $d$ is the constant with all of them being real numbers. An equation involving a cubic polynomial is called a cubic equation.

Some of the examples of a cubic polynomial are

$4x^{3}$ , $2x^{3}+1$ , $5x^{3}+x^{2}$ , $2x^{3}+4x^{2}+6x+7$

Example

Factorise $x^{3}-23x^{2}+142x-120$

$x^{3}-23x^{2}+142x-120$

$x^{3}-x^{2}-22x^{2}+22x+120x-120$

$x^{2}(x-1)-22x(x-1)+120(x-1)$

$(x-1)(x^{2}-22x+120)$ Taking $(x-1)$ common

Now factorise $x^{2}-22x+120$

$x^{2}-22x+120$

$x^{2}-10x-12x+120$

$x(x-10)-12(x-10)$

$(x-10)(x-12)$

$x^{3}-23x^{2}+142x-120=(x-1)(x-10)(x-12)$

Solution : Let p(x) = x 3 – 23x 2 + 142x – 120 We shall now look for all the factors of –120. Some of these are ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±24, ±30, ±60. By trial, we find that p(1) = 0. So x – 1 is a factor of p(x). Now we see that x 3 – 23x 2 + 142x – 120 = x 3 – x 2 – 22x 2 + 22x + 120x – 120 = x 2 (x –1) – 22x(x – 1) + 120(x – 1) (Why?) = (x – 1) (x 2 – 22x + 120) [Taking (x – 1) common] We could have also got this by dividing p(x) by x – 1. Now x 2 – 22x + 120 can be factorised either by splitting the middle term or by using the Factor theorem. By splitting the middle term, we have: x 2 – 22x + 120 = x 2 – 12x – 10x + 120 = x(x – 12) – 10(x – 12) = (x – 12) (x – 10) So, x 3 – 23x 2 – 142x – 120 = (x – 1)(x – 10)(x – 12)