Question

Factorize \[12k{y^2} + 8ky - 20k\].

Answer 1

Hint: Factorization or factoring is defined as the breaking or decomposition of an entity (for example, a number, a matrix, or a polynomial) into a product of another entity, or factors, which when multiplied together yield the original number, matrix, or polynomial, etc.
i.e.
\[
  k(a + b) = k \times a + k \times b \\
  (a + b)(c + d) = a \times c + a \times d + b \times c + b \times d \\
 \]

Complete step by step solution:
The given equation to factorize is \[12k{y^2} + 8ky - 20k\]
By using the above example method, we are going to find the solution for this given question.
Method: Splitting the Middle term
Given: \[12k{y^2} + 8ky - 20k\]
The Middle term is \[8ky\] is to be expanded as
 \[8ky = 20ky - 12ky\]
Applying the expanded value of the middle term in the given equation, then
\[ \Rightarrow 12k{y^2} + 20ky - 12ky - 20k\]
By explicating with the common terms from the above equation
\[ \Rightarrow 4ky(3y + 5) - 4k(3y + 5)\]
Combining the similar factors
\[ \Rightarrow (4ky - 4k)(3y + 5)\]
Hence the factors of the given equation \[12k{y^2} + 8ky - 20k\] are \[(4ky - 4k), (3y + 5)\].

Note:
It is simply dividing an integer or polynomial into factors that, when multiplied together, result in the original or initial integer or polynomial.
We use the factorization method to simplify any algebraic or quadratic equation by representing it as the product of factors rather than expanding the brackets.
Any equation's factors can be an integer, a variable, or the algebraic expression itself.
For Example,
Factorize: \[6{x^2} + 5x - 6\]
The method which we used to split the equation is by splitting the middle term,
Here the middle term is \[5x\]
Which we are splitting as \[5x = - 4x + 9x\].
Hence applying it in the equation of the given
\[6{x^2} + 5x - 6\]
\[ \Rightarrow 6{x^2} - 4x + 9x - 6\]
By explicating with a common factor, we got as,
\[ \Rightarrow 2x(3x - 2) + 3(3x - 2)\]
Combing the similar factors
\[ \Rightarrow (2x + 3)(3x - 2)\]
Thus, the factors of the given equation are \[(2x + 3)(3x - 2)\]