Question

Find the following products using appropriate identities.
A. $ \left( {x + 3} \right)\left( {x + 3} \right) $
B. $ \left( {x - 3} \right)\left( {x + 5} \right) $

Answer 1

Hint: Student can use this identity; $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $ which states that the square of the sum of two numbers is equal to sum of the squares of the numbers that is to be added to twice the value of the product of two numbers.

Complete step-by-step answer:
 $ \left( {x + 3} \right)\left( {x + 3} \right) $ is the given expression.
It is in the form of $ \left( {a + b} \right) \cdot \left( {a + b} \right) $ .
The product of $ \left( {a + b} \right) \cdot \left( {a + b} \right) $ can be given as
 $ \left( {a + b} \right) \cdot \left( {a + b} \right) = {\left( {a + b} \right)^2} $
There is an identity which is $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $ .
 $ \therefore \left( {a + b} \right) \cdot \left( {a + b} \right) = {a^2} + 2ab + {b^2}......\left( 1 \right) $
Compare $ \left( {x + 3} \right)\left( {x + 3} \right) $ with $ \left( {a + b} \right) \cdot \left( {a + b} \right) $ , we get $ a = x,{\rm{ }}b = 3 $
After substituting $ a = x,{\rm{ }}b = 3 $ in the equation (1), we get
 $
\Rightarrow \left( {x + 3} \right)\left( {x + 3} \right) = {x^2} + 2 \cdot x \cdot 3 + {3^2}\\
 = {x^2} + 6x + 9
 $
 $ \left( {x - 3} \right)\left( {x + 5} \right) $ is the given expression.
It is in the form of $ \left( {x - a} \right) \cdot \left( {x + b} \right) $
The product of $ \left( {x - a} \right) \cdot \left( {x + b} \right) $ can be given as
 $\Rightarrow \left( {x - a} \right) \cdot \left( {x + b} \right) = {x^2} + x\left( {b - a} \right) - ab......\left( 2 \right) $
After comparing the expressions $ \left( {x - 3} \right)\left( {x + 5} \right) $ and $ \left( {x - a} \right) \cdot \left( {x + b} \right) $ , we get $ a = 3,{\rm{ }}b = 5 $ .
Now we can substitute $ a = 3,{\rm{ }}b = 5 $ in the equation (2) to find the product of $ \left( {x - 3} \right)\left( {x + 5} \right) $ .
 $
\Rightarrow \left( {x - 3} \right)\left( {x + 5} \right) = {x^2} + x\left( {5 - 3} \right) - \left( {3 \cdot 5} \right)\\
 = {x^2} + 2x - 15
 $
Therefore, the product of $ \left( {x - 3} \right)\left( {x + 5} \right) $ is $ {x^2} + 2x - 15 $ and the product of $ \left( {x + 3} \right)\left( {x + 3} \right) $ is $ {x^2} + 6x + 9 $ .

Additional Information:
There are some algebraic equations which have only positive integral powers. Such algebraic equations are called polynomial identities. They relate two seemingly unrelated quantities with the help of variables and numbers. Many of the real-life problems have been solved with the help of polynomial identities.
There are some of the important identities which are mostly used to either factorise or find the product of an expression.
 $\Rightarrow {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $
The square of the sum of two numbers can be given as the sum of the squares of the individual numbers added to twice the product of those two numbers.
$\Rightarrow {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $
The square of the difference of two numbers can be given by subtracting twice the product of the numbers from the sum of the squares of the individual numbers.
$\Rightarrow {a^2} - {b^2} = \left( {a + b} \right) \cdot \left( {a - b} \right) $
The difference between the squares of the two numbers is equal to the product of the sum and difference of those two numbers.


Note: In this type of question, students can also open up the brackets of the given expression and multiply term by term to solve it. Students can often use the rule of BODMAS, which states that the first bracket should be solved, then division, then multiplication, addition and at the last subtraction.