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How do you factor $2{x^3} + 3{x^2} - 8x - 12$?

Last updated date: 19th Jul 2024
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Hint: We are given the polynomial. We have to factor the polynomial. First, we will do a grouping of terms which have a common factor. Then, we will take out the common terms. Then, we will apply the algebraic identity to the expression to further factorize the expression. Then, we will write the factors.

Complete step by step solution:
We are given the polynomial, $2{x^3} + 3{x^2} - 8x - 12$
Now, write the expression by grouping the terms which have a common factor.
$ \Rightarrow \left( {2{x^3} + 3{x^2}} \right) + \left( { - 8x - 12} \right)$
Now, move out the common factor from each group.
$ \Rightarrow {x^2}\left( {2x + 3} \right) - 4\left( {2x + 3} \right)$
Now, again move out the common factor $\left( {2x + 3} \right)$from the expression.
$ \Rightarrow \left( {2x + 3} \right)\left( {{x^2} - 4} \right)$
Now, factorize the expression $\left( {{x^2} - 4} \right)$ completely by applying the algebraic identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$
Here, $a = x$ and $b = 2$. Use the algebraic identity, we get:
$ \Rightarrow {\left( x \right)^2} - {\left( 2 \right)^2} = \left( {x - 2} \right)\left( {x + 2} \right)$
Now, write the factors of the expression.
$ \Rightarrow \left( {2x + 3} \right)\left( {x - 2} \right)\left( {x + 2} \right)$

Final Answer: Thus, the factors of the expression, $2{x^3} + 3{x^2} - 8x - 12$ are $\left( {2x + 3} \right)\left( {x - 2} \right)\left( {x + 2} \right)$

Additional Information:
The factorization of the polynomial is done to reduce the polynomial in the simplified form. In the factorization process, the terms which are adding or subtracting are converted into the product of terms form. In factorization, the higher degree polynomials are converted into linear factors. The polynomials can be factorized in many different ways: the first method is by splitting the middle term of the polynomial, the second method is by grouping. In this grouping, method pairs are formed by grouping the first and second term together and the third, fourth term together. Another method to factorize the polynomial is by the perfect squares method.

In such types of questions students mainly make mistakes while making groups of the terms. In such types of questions, the polynomial is factored to such an extent that it cannot be further simplified.