# Factorization of polynomial – set 01

Given below is collection of questions related to factorization of polynomial.

All the questions are provided with detailed solution.

The questions are to the standard of grade 09 math.

Question 01
Find the value of given polynomial for f (2) and f (-3).
f(x)= \mathtt{2x^{3} -13x^{2} +17x+12}

Solution
Let’s first calculate the value of f(2).

Putting the value of 2 in given function.

\mathtt{\Longrightarrow \ 2( 2)^{3} -13( 2)^{2} +17( 2) +12}\\\ \\ \mathtt{\Longrightarrow \ 2( 8) -13( 4) +34+12}\\\ \\ \mathtt{\Longrightarrow \ 16-52+34+12}\\\ \\ \mathtt{\Longrightarrow \ 10}

Hence, value of f(2) is equal to 10.

Now let’s calculate the value of f(-3)

Putting the value of -3 in the given function.

\mathtt{\Longrightarrow \ 2( -3)^{3} -13( -3)^{2} +17( -3) +12}\\\ \\ \mathtt{\Longrightarrow \ 2( -27) -13( 9) -51+12}\\\ \\ \mathtt{\Longrightarrow \ -54-117-51\ +\ 12}\\\ \\ \mathtt{\Longrightarrow \ -210}

Hence, value of f(-3) is equal to -210.

Question 02
It’s given that x =2 is the root of following polynomial f(x).
f(x)= \mathtt{2x^{2} -3x+7a}

Find the value of “a” in given function.

Solution
Since x = 2 is the root of given function.
The value of function f(x) will be zero when we put 2 in the given expression.

f(2) = 0

\mathtt{2( 2)^{2} -3( 2) +7a=0}\\\ \\ \mathtt{2( 4) -6+7a=\ 0}\\\ \\ \mathtt{8-6+7a\ =\ 0}\\\ \\ \mathtt{7a\ =\ -2}\\\ \\ \mathtt{a=-2/7}

Hence, value of a is -2/7

Question 03
if x = -1/2 is the root of below polynomial f(x).
f(x) = \mathtt{8x^{3} -ax^{2} -x+2}

Find the value of variable “a”.

Solution
Since x = -1/2 is the root of given expression.

Putting the value of x =-1/2 will result in 0.

\mathtt{8\left( -\frac{1}{2}\right)^{3} -a\left(\frac{-1}{2}\right)^{2} -\left(\frac{-1}{2}\right) +2=0}\\\ \\ \mathtt{8\left(\frac{-1}{8}\right) -a\left(\frac{1}{4}\right) +\frac{1}{2} +2=0}\\\ \\ \mathtt{-1-\frac{a}{4} +\frac{5}{2} =0}\\\ \\ \mathtt{\frac{-4-a+10}{4} =0}\\\ \\ \mathtt{-4-a+10=0}\\\ \\ \mathtt{a=6}

The value of a is 6.

Question 04
x= 0 and x = -1 are the root of following polynomial f(x).
f(x) = \mathtt{2x^{3} -3x^{2} +ax+b}

Find the value of a and b.

Solution
Since x = 0 is the root, the value of function is 0.

So f(0) = 0

\mathtt{2( 0) -3( 0) +a( 0) +b=0}\\\ \\ \mathtt{b=0}

Hence, value of b = 0.

Also x= -1 is also the root.
f(-1) = 0

\mathtt{2( -1)^{3} -3( -1)^{2} +a( -1) +0=0}\\\ \\ \mathtt{-2-3-a=0}\\\ \\ \mathtt{a\ =-5}

Hence, the value of a = -5.

Question 05
Check if x = 1 is the root of below function f(x).
f (x) = \mathtt{\ 9x^{3} -5x^{2} -7x+3}

Solution
Put x = 1 in the given function.

\mathtt{\Longrightarrow \ 9( 1)^{3} -5( 1)^{2} -7( 1) +3}\\\ \\ \mathtt{\Longrightarrow \ 9-5-7+3}\\\ \\ \mathtt{\Longrightarrow \ 12-12}\\\ \\ \mathtt{\Longrightarrow \ 0}

Since value of function gets 0, number 1 is the root of the given function.

Next chapter : Problems on factorization using factor theorem