Question

How do you factor $2{{x}^{2}}+9x+10$ ?

Answer 1

Hint: The given expression has three terms that is an odd number of terms. Whenever there are an odd number of terms, we split the middle term which is 9x using the sum and product of the coefficients of the end terms. After splitting we get an even number of terms. Likewise put the factors out from the first two terms and then the next two terms. Write them all together and represent them as factors.

Complete step by step solution:
The given polynomial which must be factorized is $2{{x}^{2}}+9x+10$
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 $a=2;b=9;c=10$
The product of the terms is $2\times 10=20$
The sum of the terms is $9$
Therefore, the terms can be $4x,5x$
On substituting back, we get,
$\Rightarrow 2{{x}^{2}}+4x+5x+10$
The polynomial is of degree $2$
Now, firstly let us take the common terms out of the first two terms.
$\Rightarrow 2x\left( x+2 \right)+5x+10$
Now secondly take the common terms out of the last two terms.
$\Rightarrow 2x\left( x+2 \right)+5\left( x+2 \right)$
Now on writing it in the form of the product of sums, also known as factoring,
$\Rightarrow \left( 2x+5 \right)\left( x+2 \right)$
Now writing all the factors together we get,
$\Rightarrow \left( 2x+5 \right)\left( x+2 \right)$

Hence the factors for the polynomial $2{{x}^{2}}+9x+10$ are $\left( 2x+5 \right)\left( x+2 \right)$

Note: We can verify our solution by cross multiplying our factors to get the question as a solution.
The factors are $\left( 2x+5 \right)\left( x+2 \right)$
On multiplying them we get,
$\Rightarrow \left( 2x+5 \right)\left( x+2 \right)$
$\Rightarrow \left( 2x\left( x+2 \right)+5\left( x+2 \right) \right)$
On opening the brackets and then evaluating we get,
$\Rightarrow \left( 2{{x}^{2}}+4x+5x+10 \right)$
$\Rightarrow \left( 2{{x}^{2}}+9x+10 \right)$
Since we got the question back as our solution, the factors are correct.