Question

Factorize the given quadratic polynomial: $ 2{{x}^{2}}-18 $ ?

Answer 1

Hint:
We start solving the problem by adding and subtracting the term $ 6x $ to the given quadratic polynomial $ 2{{x}^{2}}-18 $ . We then make the necessary arrangements and make use of the distributive property $ a\left( b+c \right)=ab+ac $ to proceed through the problem. We then make use of the distributive property $ \left( a+b \right)c=ac+bc $ to get the factorization of the given quadratic polynomial which is the required answer.

Complete step by step answer:
According to the problem, we are asked to factorize the given quadratic polynomial: $ 2{{x}^{2}}-18 $.
Now, we have the polynomial $ 2{{x}^{2}}-18 $ .
Let us add and subtract the given polynomial with the term $ 6x $ .
 $ \Rightarrow 2{{x}^{2}}-18=2{{x}^{2}}+6x-6x-18 $ .
 $ \Rightarrow 2{{x}^{2}}-18=\left( 2x\times x \right)+\left( 2x\times 3 \right)+\left( -6\times x \right)+\left( -6\times 3 \right) $ ---(1).
From distributive property, we know that $ a\left( b+c \right)=ab+ac $ . Let us use this result in equation (1).
 $ \Rightarrow 2{{x}^{2}}-18=2x\left( x+3 \right)-6\left( x+3 \right) $ ---(2).
From distributive property, we know that $ \left( a+b \right)c=ac+bc $ . Let us use this result in equation (2).
 $ \Rightarrow 2{{x}^{2}}-18=\left( 2x-6 \right)\left( x+3 \right) $ .
So, we have found the factorization of the quadratic polynomial $ 2{{x}^{2}}-18 $ as $ \left( 2x-6 \right)\left( x+3 \right) $ .
 $ \therefore $ The factors of the quadratic polynomial $ 2{{x}^{2}}-18 $ is $ \left( 2x-6 \right)\left( x+3 \right) $ .

Note:
 We can also solve this problem by first finding the zeroes of the polynomial and then making use of the fact that the factors of the polynomial with zeroes a, b is $ \left( x-a \right) $ , $ \left( x-b \right) $ . We should keep in mind that if we multiply the obtained factors, we should get a similar polynomial as the result. We should not report the result of factorization as $ \left( x-3 \right)\left( x+3 \right) $ by dividing 2 from the factor which is the common mistake done by students. Similarly, we can expect problems to find the zeroes of the given quadratic polynomial.