Question

How do you factor $-2{{x}^{2}}+7x+1=0$ ?

Answer 1

Hint: The number of terms in the given expression is three. To take common factors out we require a minimum of even number of terms. Hence, we shall split the middle term 7x. Sometimes we do not find any way to split the term. That is when we do not get any factors for the expression. But in our question, we can take the negative symbol out and consider that as a factor.

Complete step by step solution:
The given polynomial which must be factorized is $-2{{x}^{2}}+7x+1=0$
First, let us try to take any common integer or variable from all three terms.
Since there are no common terms in all three terms,
To factorize this polynomial, we must first split the middle term.
For this, we use the product sum form.
If the polynomial’s general form is $a{{x}^{2}}+bx+c=0$
Then to split the middle term, we use the product and sum formula.
Which is,
The product of the middle terms after splitting must be $a\times c$
And the sum of the middle terms must be $b$
Here $a=-2;b=7;c=1$
The product of the terms is $-2\times 1=-2$
The sum of the terms is $7$
As we can see that we are not able to split the middle term.
The only factor for this polynomial will be $-1$
Now writing the factors together we get,
$\Rightarrow \left( -1 \right)\left( 2{{x}^{2}}-7x-1 \right)$

Hence the factors for the polynomial $-2{{x}^{2}}+7x+1=0$ are $\left( -1 \right)\left( 2{{x}^{2}}-7x-1 \right)$.

Note: For some polynomials when we try to take common factors out or try to split the middle term, we do not get the results, or we find it difficult to split. These polynomials do not have any factor other than one and itself. It is the same as the prime numbers. They do not have other factors rather than one and itself.