Factorization of algebraic expression – Problem set 02

Given below is collection of question related to factorization of algebraic expression.

All the questions are to the standard of grade 09 math.

Question 01
Factorize the below expression;
\mathtt{a^{2} +b^{2} +2( ab+bc+ca)}

Solution
\mathtt{\Longrightarrow \ a^{2} +b^{2} +2( ab+bc+ca)}\\\ \\ \mathtt{\Longrightarrow \ a^{2} +b^{2} +2ab\ +2bc+2ca}\\\ \\ \mathtt{\Longrightarrow \ ( a+b)^{2} +2bc+2ca}\\\ \\ \mathtt{\Longrightarrow \ ( a+b)^{2} +2c( a+b)}\\\ \\ \mathtt{\Longrightarrow \ ( a+b)( a+b+2c)}

Hence, the above expression is the solution.

Question 02
Factorize the below expression:
\mathtt{xy^{9} -yx^{9}}

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

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

Using the formula in above expression;

\mathtt{\Longrightarrow xy\left(\left( y^{4}\right)^{2} -\left( x^{4}\right)^{2}\right) \ }\\\ \\ \mathtt{\Longrightarrow \ xy\left( y^{4} -x^{4}\right)\left( y^{4} +x^{4}\right)}\\\ \\ \mathtt{\Longrightarrow \ xy\left(\left( y^{2}\right)^{2} -\left( x^{2}\right)^{2}\right)\left( y^{4} +x^{4}\right)}\\\ \\ \mathtt{\Longrightarrow \ xy\ \left( y^{2} -x^{2}\right)\left( y^{2} +x^{2}\right)\left( y^{4} +x^{4}\right)}\\\ \\ \mathtt{\Longrightarrow \ xy\ ( y-x)( y+x)\left( y^{2} +x^{2}\right)\left( y^{4} +x^{4}\right)}

Hence, the above expression is the solution.

Question 03
Factorize the below algebraic expression.
\mathtt{x^{4} +x^{2} y^{2} +y^{4}}

Solution
Add & subtract \mathtt{x^{2} y^{2}} in the given expession;

\mathtt{\Longrightarrow \ x^{4} +x^{2} y^{2} +y^{4} +x^{2} y^{2} -x^{2} y^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{4} +2x^{2} y^{2} +y^{4} -x^{2} y^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2}\right)^{2} +2x^{2} y^{2} +\left( y^{2}\right)^{2} -x^{2} y^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} +y^{2}\right)^{2} -( xy)^{2}}

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

Using the formula, we get;
\mathtt{\Longrightarrow \ \left( x^{2} +y^{2}\right)^{2} -( xy)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} +y^{2} +xy\right)\left( x^{2} +y^{2} -xy\right)}

Hence, the above expression is the solution.

Question 04
Factorize the below expression;
\mathtt{x^{3} +x-3x^{2} -3}

Solution
Factorizing 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 solution.

Question 05
Find the possible length & breadth of rectangle if area is given in form of expression;
\mathtt{35x^{2} +28x-15x-12}

Solution
Factorizing the expression;

\mathtt{\Longrightarrow \ 35x^{2} +28x-15x-12}\\\ \\ \mathtt{\Longrightarrow \ 7x( 5x+4) -3( 5x+4)}\\\ \\ \mathtt{\Longrightarrow \ ( 5x+4)( 7x-3)}

We know that;
Area of rectangle = Length x Breadth

On comparison with above expression;
Length = 5x + 4
Breadth = 7x – 3 or vice-versa

Next chapter: Problems on factorization – set 03

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