Question

Factors of $8{{x}^{2}}-18x+9$ are:
A. $\left( 4x-3 \right)and\left( 2x+3 \right)$
B. $\left( 8x-1 \right)and\left( x-9 \right)$
C. $\left( 8x-3 \right)and\left( x-3 \right)$
D. $\left( 2x-3 \right)and\left( 4x-3 \right)$

Answer 1

Hint: To obtain the factors of the given polynomial we will use splitting the middle term method. Firstly we will check what is the product of the coefficient of ${{x}^{2}}$ and the constant. Then according to the product obtained we will split the middle term such that the product of the two coefficients obtained after splitting is equal to the product of coefficients of ${{x}^{2}}$ and the constant. Finally we will take common from first two and last two terms and simplify it to get our desired answer.

Complete step-by-step answer:
We have the factors of the below quadratic polynomial:
$8{{x}^{2}}-18x+9$….$\left( 1 \right)$
Now we will find the product of coefficient of ${{x}^{2}}$ and the constant as below:
$\begin{align}
  & = 8\times 9 \\
 & = 72 \\
\end{align}$
So we will split the middle term so that the product of the coefficient of the two term obtain is 72 and the sum of them give us the middle term as below:
$\begin{align}
  & 8{{x}^{2}}-18x+9 \\
 & = 8{{x}^{2}}-\left( 12+6 \right)x+9 \\
 & = 8{{x}^{2}}-12x-6x+9 \\
\end{align}$
 Next we will simplify the polynomial further to get the two factors as below:
$= 4x\left( 2x-3 \right)-3\left( 2x-3 \right)$
$= \left( 2x-3 \right)\left( 4x-3 \right)$
So we get the factors as $\left( 2x-3 \right)\left( 4x-3 \right)$

So, the correct answer is “Option D”.

Note: Quadratic polynomials are those polynomials which have variables with highest power as 2. We can also solve the given polynomial by using factorization method but as we have a small coefficient using splitting the middle method is easier. When the coefficients are large then using this method can become complicated as we have to be extra careful while multiplying the coefficient or finding the right term to split the middle term.