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Unit 9: Factorization

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Class 9: Mathematics

Factorization, Important Formulae, Example Questions with Solutions, Factorization of the expression in the form of \(a^4 + a^2b^2 + b^4\), Example Qu...

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    Factorization

    The process of expressing a polynomial expression as the product of its factor is called factorization. In this case, the part of the expression is called the factor of the polynomial. 

    Example:

    If the polynomial \(x^2 + 5x + 6\) is expressed as \((x+3) (x+2)\) then \((x+3) \text{ and } (x+2)\) are the factors of \(x^2 + 5x + 6\)

     

    Important Formulae

    Here are the important formulae that can be used while solving the problems.

    \( \begin{aligned} & \textbf{Formulae} \\ & 1. (a + b)^2 = a^2 + 2ab + b^2 \\ & 2. (a - b)^2 = a^2 - 2ab + b^2 \\ & 3. (a^2 + b^2) = (a + b)^2 – 2ab \text{ or, } (a - b)^2 + 2ab \\ & 4. (a^2 – b^2) = (a + b)(a - b) \\ & 5. (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\ & 6. (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \\ & 7. (a^3 + b^3) = (a + b)(a^2 - ab + b^2) \\ & 8. (a^3 - b^3) = (a - b)(a^2 + ab + b^2) \\ \end{aligned} \)

     

    Example Questions with Solutions

    Tips and Tricks

    Some Tips and Tricks for Factorizing the given expressions.

    1. Look for a common factor.
    2. Analyze the expression to see if we can use formulae.
    3. Check your steps thoroughly to see if anything went wrong.

     

    Example Questions with Solutions

    Here are some example questions.

    \( \begin{aligned} & \textbf{ Q1. Factorize } \mathbf{(x + 2)^3} \\ & \text{Solution:} \\ & = (x + 2)^3 \\ & = x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2 + 2^3\\ & \therefore x^3 + 6x^2 + 12x + 8 \end{aligned} \)
    \( \begin{aligned} & \textbf{ Q2. Factorize } 8x^4 + 27x \\ & \text{Solution:} \\ & = 8x^4 + 27x \\ & = x(8x^3 + 27) \\ & = x\lbrace(2x)^3 + 3^3\rbrace \\ & = x[(2x + 3)\lbrace(2x)^2 + 2x \cdot 3 + 3^2\rbrace] \\ & \therefore x(2x + 3)(4x^2 + 6x + 9) \\ \end{aligned} \)
    \( \begin{aligned} & \textbf{ Q3. Factorize } (\frac{x^3}{y^3} - \frac{y^3}{x^3}) \\ & \text{Solution:} \\ & = \frac{x^3}{y^3} - \frac{y^3}{x^3} \\ & = (\frac {x}{y})^3 – (\frac{y}{x})^3 \\ & = (\frac{x}{y} - \frac{y}{x}) \lbrace (\frac{x}{y})^2 + \frac{x}{y} \cdot \frac{y}{x} + (\frac{y}{x})^2\rbrace \\ & \therefore (\frac{x}{y} - \frac{y}{x}) (\frac{x^2}{y^2} + \frac{y^2}{x^2} + 1) \\ \end{aligned} \)

     

    Factorization of the expression in the form

    \((a^4 + a^2b^2 + b^4)\)

    Here are the general steps for the factorization of the expression in the form of \((a^4 + a^2b^2 + b^4)\).

    \( \begin{aligned} & \text{Steps} \\ & = a^4 + a^2b^2 + b^4 \\ & = (a^2)^2 + (b^2)^2 + a^2b^2\\ & = (a^2 + b^2)^2 – 2a^2b^2 + a^2b^2\\ & = (a^2 + b^2)^2 – (ab)^2\\ & \therefore (a^2 + ab + b^2)( a^2 - ab + b^2) \end{aligned} \)

     

    Example Questions with Solution

    Here are some example questions and solutions.

    \( \begin{aligned} & \text{Q1. Factorize } (x^4 + x^2y^2 + y^4) \\ & \text{Solution:} \\ & = x^4 + x^2y^2 + y^4 \\ & = (x^2)^2 + (y^2)^2 + x^2y^2\\ & = (x^2 + y^2)^2 – 2x^2y^2 + x^2y^2\\ & = (x^2 + y^2)^2 – (xy)^2\\ & \therefore (x^2 + xy + y^2)( x^2 - xy + y^2) \end{aligned} \)
    \( \begin{aligned} & \text{Q2. Factorize } (x^4 + 5x^2y^2 + 4y^4) \\ & \text{Solution:} \\ & = x^4 - 5x^2y^2 + 4y^4 \\ & = (x^2)^2 + (2y^2)^2 - 5x^2y^2 \\ & = (x^2 + 2y^2)^2 – 2 \cdot x^2 \cdot 2y^2 - 5x^2y^2 \\ & = (x^2 + 2y^2)^2 – 4x^2y^2 - 5x^2y^2 \\ & = (x^2 + 2y^2)^2 – 9x^2y^2 \\ & = (x^2 + 2y^2)^2 – (3xy)^2 \\ & \therefore (x^2 + 3xy + 2y^2)( x^2 - 3xy + 2y^2) \end{aligned} \)
    \( \begin{aligned} & \text{Q3. Factorize } (y^4 + \frac{1}{y^4} + 1) \\ & \text{Solution:} \\ & = y^4 + \frac{1}{y^4} + 1 \\ & = (y^2)^2 + (\frac{1}{y^2})^2 + 1 \\ & = (y^2 + \frac{1}{y^2})^2 – 2 \cdot y^2 \cdot \frac{1}{y^2} + 1 \\ & = (y^2 + \frac{1}{y^2})^2 – 2 + 1 \\ & = (y^2 + \frac{1}{y^2})^2 – 1^2 \\ & \therefore (y^2 + \frac{1}{y^2} + 1)(y^2 + \frac{1}{y^2} - 1) \end{aligned} \)

     

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