Question

Factorize the expressions and divide them as directed \[12xy\left( {9{x^2} - 16{y^2}} \right) \div 4xy\left( {3x + 4y} \right)\].
A) \[3\left( {3x - 4y} \right)\]
B) \[3\left( {3x + 4y} \right)\]
C) \[4\left( {3x - 4y} \right)\]
D) \[3\left( {4x - 3y} \right)\]

Answer 1

Hint: The process of finding factors of a given value is called factorisation. Factors are the integers which are multiplied to produce an original number. To factorize the given expression, solve with respect to numerator and denominator terms, hence simplifying it we can get the required expression.

Complete step-by-step solution:
 The given expressions are
\[12xy\left( {9{x^2} - 16{y^2}} \right) \div 4xy\left( {3x + 4y} \right)\]
The expressions can be written as
\[\dfrac{{12xy\left( {9{x^2} - 16{y^2}} \right)}}{{4xy\left( {3x + 4y} \right)}}\] ………….. 1
The numerator and denominator terms can be expressed as
\[12xy\left( {9{x^2} - 16{y^2}} \right) = 12xy{\left( {3x} \right)^2}\]
\[12xy\left( {9{x^2} - 16{y^2}} \right) = 12xy\left( {3x + 4y} \right)\left( {3x - 4y} \right)\]
From equation 1 we get
= \[\dfrac{{12xy\left( {3x + 4y} \right)\left( {3x - 4y} \right)}}{{4xy\left( {3x + 4y} \right)}}\]
= \[\dfrac{{3\left( {3x + 4y} \right)\left( {3x - 4y} \right)}}{{\left( {3x + 4y} \right)}}\]
Simplifying the terms, we get
= \[3\left( {3x - 4y} \right)\]
Therefore,
\[\dfrac{{12xy\left( {9{x^2} - 16{y^2}} \right)}}{{4xy\left( {3x + 4y} \right)}}\] = \[3\left( {3x - 4y} \right)\]

Hence, option A is the correct answer.

Additional information: Types of Factoring polynomials
Greatest Common Factor (GCF): We have to find out the greatest common factor, of the given polynomial to factorise it.
Grouping Method: This method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros.
Factoring Using Identities: The factorisation can be done also by using algebraic identities.
Let us know the Factor theorem:
For a polynomial \[p\left( x \right)\] of degree greater than or equal to one: x-a is a factor of \[p\left( x \right)\], if \[p\left( a \right) = 0\] and if \[p\left( a \right) = 0\], then x-a is a factor of \[p\left( x \right)\].
Where ‘a’ is a real number.


Note: The key point to evaluate any trigonometric function is that we must know all the basic trigonometric functions and their relation. As in the given equation consists of cot functions, hence we must know all the trigonometric identities with respect to the function, as we know that \[\tan \left( {{{90}^\circ } - \theta } \right) = \cot \theta \]hence, by applying this we can evaluate the given functions.