In this chapter we will discuss the concept of factor theorem of polynomial and will also solve some questions related to it.
Let’s start with the basics.
What is factor of polynomial ?
The expression which completely divides the polynomial without leaving any remainder is called factor of polynomial.
For example, consider the following polynomial f(x).
f(x) = \mathtt{x^{2} +2x+1}
Here (x + 1) is the factor of above polynomial f(x) as if we divide f(x) by (x + 1), the expression will be fully divisible.
Let’s check the division.
\mathtt{\Longrightarrow \ \frac{\mathtt{x^{2} +2x+1}}{x+1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( x+1)^{2}}{x+1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{( x+1)}( x+1)}{\cancel{x+1}}}
You can see that the denominator completely divides numerator without leaving any remainder.
Hence, (x + 1) is factor of polynomial f(x).
What is factor theorem of polynomial ?
Let us consider two polynomial f(x) and g(x).
If we put root of g(x) on f(x) and get 0 as a result, it means that g(x) is the factor of polynomial f(x).
Let us understand the theorem with the help of examples.
Question 01
Consider the two below polynomials.
f(x) = \mathtt{x^{3} +3x^{2} +5x+6}
g(x) = x + 2
Check if g(x) is the factor of polynomial f(x).
Solution
We will solve the question using factor theorem of polynomials.
Lets first find the root of g(x).
x + 2 = 0
x = -2
Hence, -2 is the root of function g(x).
Put this root on polynomial f(x). If we get 0 then g(x) is the factor of f(x).
\mathtt{\Longrightarrow \ ( -2)^{3} +3( -2)^{2} +5( -2) +6\ }\\\ \\ \mathtt{\Longrightarrow \ \ -8+12-10+6}\\\ \\ \mathtt{\Longrightarrow \ 18\ -\ 18}\\\ \\ \mathtt{\Longrightarrow \ 0}
Hence. g(x) is the factor of f(x).
Question 02
Consider the two polynomials f(x) and g(x).
f(x)= \mathtt{x^{3} \ –\ 4x^{2} \ +\ x\ +\ 6}
g(x) = x – 3
Check if g(x) is the factor of f(x).
Solution
Let’s first find the root of g(x)
x – 3 = 0
x = 3
Put this value of x in polynomial f(x).
\mathtt{\Longrightarrow \ 3^{3} \ –\ 4( 3)^{2} \ +\ 3\ +\ 6}\\\ \\ \mathtt{\Longrightarrow \ 27-36+\ 9}\\\ \\ \mathtt{\Longrightarrow \ 0}
Since we get 0, it means that g(x) is the factor of polynomial f(x).
Question 03
Consider the two functions f(x) and g(x).
f(x) = \mathtt{2x^{3} +x^{2} \ –\ 2x\ –\ 1}
g(x) = x +1
Check if g(x) is the factor of polynomial f(x).
Solution
Let’s first find the root of g(x).
x + 1 = 0
x = -1
Put this value of x in polynomial f(x).
\mathtt{\Longrightarrow \ 2( -1)^{3} +( -1)^{2} \ –\ 2( -1) \ –\ 1}\\\ \\ \mathtt{\Longrightarrow \ -2+1+2-1}\\\ \\ \mathtt{\Longrightarrow \ 0}
Since we are getting 0, it tells that g(x) is factor of f(x).
Next chapter : Problems on factorization of polynomials