Question

How do you factor $3{y^4} - 2{y^2} - 5$ completely?

Answer 1

Hint: To solve this question, we will assume$x = {y^2}$. Now, this equation is the quadratic equation. The general form of the quadratic equation is$a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term. To solve this equation, we will apply the sum-product pattern. During the simplification, we will take out common factors from the two pairs. Then we will rewrite it in factored form.
 Therefore, we should follow the below steps:
> Apply sum-product pattern.
> Make two pairs.
> Common factor from two pairs.
> Rewrite in factored form.

Complete step-by-step answer:
In this question, we want to factor in the given expression.
The given expression is,
$ \Rightarrow 3{y^4} - 2{y^2} - 5$
In the above expression, we can write the term ${y^4}$as ${\left( {{y^2}} \right)^2}$ .
Let us substitute ${y^4}$term in the above expression.
 $ \Rightarrow 3{\left( {{y^2}} \right)^2} - 2{y^2} - 5$
Now, let us assume$x = {y^2}$.
Substitute x in the above expression instead of ${y^2}$.
$ \Rightarrow 3{x^2} - 2x - 5$
This equation is the quadratic equation.
Let us apply the sum-product pattern in the above equation.
Since the coefficient of ${x^2}$is 3 and the constant term is -5. Let us multiply 3 and -5. The answer will be -15. We have to find the factors of 15 which sum to -2. Here, the factors are -5 and 3.
Therefore,
$ \Rightarrow 3{x^2} - 5x + 3x - 5$
Now, make two pairs in the above equation.
$ \Rightarrow \left( {3{x^2} - 5x} \right) + \left( {3x - 5} \right)$
Let us take out the common factor.
$ \Rightarrow x\left( {3x - 5} \right) + 1\left( {3x - 5} \right)$
Now, rewrite the above equation in factored form.
$ \Rightarrow \left( {3x - 5} \right)\left( {x + 1} \right)$
Now, substitute ${y^2}$ in the above expression instead of x.
That is equal to,
$ \Rightarrow \left( {3{y^2} - 5} \right)\left( {{y^2} + 1} \right)$

Hence, the factors are $\left( {3{y^2} - 5} \right)$and $\left( {{y^2} + 1} \right)$.

Note:
One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
To check our factorization, multiplication goes like this:
$ \Rightarrow \left( {3x - 5} \right)\left( {x + 1} \right)$
Let us apply multiplication to remove brackets.
$ \Rightarrow 3{x^2} - 5x + 3x - 5$
Let us simplify it. We will get,
$ \Rightarrow 3{x^2} - 2x - 5$
Hence, we get our quadratic equation back by applying multiplication.