Question

Use factor theorem to factorize the polynomial $x^3-13x-12$.

Answer 1

Hint: According to Factor Theorem, zeroes and factors of polynomial are linked to each other.
Cubic polynomial is factorized by finding one factor by hit-and trial method and finding remaining two factors by factorizing the remaining quadratic polynomial.

Complete step by step solution:
To find the factors or zeroes of a cubic polynomial, firstly a factor is determined by determining the factors of constant number in the given cubic polynomial. To find factors of given cubic polynomial, factors of constant number, -12 have to be determined.
Factors of number $-12$ are: $1,-1,2,-2,3,-3,4,-4,6,-6,12,-12$.
By putting values of different factors of $-12$ into cubic polynomials, the factor of polynomial p(x) = $x^3-13x-12$ has to be determined.
At x = $4$, the value of polynomial p(x) is equal to zero. This indicates that x = $4$ is one of the zeroes of polynomials.
Now p(x) is to be divided by $4$ to determine other factors.
$$\begin{array}{c}x-4\overline{)\begin{array}{rl}& x^3-13x-12\\ & \underset{―}{-x^3+4x^2}\end{array}}(x^2+4x+3)\\ 0+4x^2-13x-12\\ \underset{―}{-4x^2+16x}\\ 3x-12\\ \underset{―}{3x-12}\\ \underset{―}{0}\end{array}$$
Further, the polynomial $(x^2+4x+3)$ has to be factorized to get the factors.
$\begin{array}{rl}& x^2+4x+3=x^2+3x+x+3\\ & =x(x+3)+1(x+3)\\ & =(x+3)(x+1)\end{array}$
When (x + $3$) (x + $1$) is set equal to zero, the zeroes obtained are:
x = $-3$, x = $-1$
This indicates that the factorization of cubic polynomial, p(x) is denoted as:
$p(x)=(x-4)(x+1)(x+3)$
Therefore, (x + $3$)(x + $1$) (x – $4$) are the factors of the given polynomial.

Note:
> Factors or zeroes of a polynomial are those numbers which when put in the polynomial make the value of the polynomial equal to zero.
> Factor theorem is used to find factors of different kinds of polynomials.