Factorization of algebraic expression – Problem set 01

In this chapter we will discuss questions related to factorization of algebraic expressions with detailed solutions.

All the questions are to the standard of grade 09.

Question 01
Factorize the below expression;
\mathtt{x^{3} +x-3x^{2} -3}

Solution
Solving the expression;

\mathtt{\Longrightarrow \ x^{3} +x-3x^{2} -3}\\\ \\ \mathtt{\Longrightarrow x\left( x^{2} +1\right) -3\left( x^{2} +1\right)}\\\ \\ \mathtt{\ \Longrightarrow \ \left( x^{2} +1\right)( x-3)}

Hence, the above expression is the factorized form.

Question 02
Factorize the below expression;
\mathtt{x\left( x^{3} -y^{3}\right) +3xy( x-y)}

Solution
Referring the formula;
\mathtt{a^{3} -b^{3} =( a-b)\left( a^{2} +ab+b^{2}\right)}

Using the formula in given expression;

\mathtt{\Longrightarrow \ x\left( x^{3} -y^{3}\right) +3xy( x-y)}\\\ \\ \mathtt{\Longrightarrow x( x-y)\left( x^{2} +xy+y^{2}\right) +3xy( x-y)}

Separating the common terms x.(x – y) from above expression, we get following;

\mathtt{\Longrightarrow \ x( x-y)\left( x^{2} +xy+y^{2} +3y\right)}

Hence, the above expression is the solution.

Question 03
Factorize the below expression;
\mathtt{\ x^{2} +y-xy-x}

Solution
Rearranging the elements;

\mathtt{\Longrightarrow \ x^{2} -xy+y-x}\\\ \\ \mathtt{\Longrightarrow \ x( x-y) +y-x}\\\ \\ \mathtt{\Longrightarrow \ -x( -x+y) +y-x}\\\ \\ \mathtt{\Longrightarrow \ -x( y-x) +y-x}\\\ \\ \mathtt{\Longrightarrow \ ( y-x)( -x+1)}

Hence, the above expression is the factorized form.

Question 04
Factorize the below algebraic expression;
\mathtt{6ab-b^{2} +12ac-2bc}

Solution
\mathtt{\Longrightarrow \ 6ab-b^{2} +12ac-2bc}\\\ \\ \mathtt{\Longrightarrow \ b( 6a-b) +2c( 6a-b)}

(6a – b) is common in both terms.

\mathtt{\Longrightarrow \ b( 6a-b) +2c( 6a-b)}\\\ \\ \mathtt{\Longrightarrow \ ( 6a-b)( b+2c)}

Hence, the above expression is the solution.

Question 05
Factorize the below algebraic expression.
\mathtt{x( x-2)( x-4) +4x-8}

Solution
\mathtt{\Longrightarrow \ x( x-2)( x-4) +4x-8}\\\ \\ \mathtt{\Longrightarrow \ x( x-2)( x-4) +4( x-2)}\\\ \\ \mathtt{\Longrightarrow \ ( x-2)( x( x-4) +4)}\\\ \\ \mathtt{\Longrightarrow \ ( x-2)\left( x^{2} -4x+4\right)}

Referring the formula;
\mathtt{( a-b)^{2} =a^{2} -ab+b^{2}}

Using the formula in above expression we get;

\mathtt{\Longrightarrow \ ( x-2)( x-2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( x-2)^{3}}

Hence, the above expression is the solution.

Question 06
Factorize the below expression;
\mathtt{x^{2} +2xy\ +\ y^{2} -z^{2}}

Solution
\mathtt{\Longrightarrow \ x^{2} +2xy\ +\ y^{2} -z^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( x+y)^{2} -z^{2}}

Referring to formula;
\mathtt{a^{2} -b^{2} =( a-b)( a+b)}

\mathtt{\Longrightarrow \ ( x+y+z)( x+y-z)}

Hence, the above expression is the solution.

Next chapter : Problems on factorization – set 02