Question

How do you write \[\left( {x - 4} \right)\left( {x + 4} \right)\] in standard form?

Answer 1

Hint: This problem is related to basic expansion formulas. These are used in algebraic expression frequently. Here we are given terms of the form \[\left( {a - b} \right)\left( {a + b} \right)\] . On solving the brackets we get \[{a^2} - {b^2}\] . This is the standard form. Furthermore if we are given the values of a and b in numbers we will find their squares and perform the subtraction.

Complete step-by-step answer:
Given that \[\left( {x - 4} \right)\left( {x + 4} \right)\]
We will solve the brackets step by step manner.
Initially multiply the first term of first bracket \[\left( x \right)\] with both the terms of second bracket and then multiply second term of first bracket \[\left( { - 4} \right)\] with both terms of second bracket.
 \[ \Rightarrow x\left( {x + 4} \right) - 4\left( {x + 4} \right)\]
Now perform the multiplication,
 \[ \Rightarrow x\times x + 4x - 4x - 4 \times 4\]
Now multiplication of term with the same terms is nothing but square,
 \[ \Rightarrow {x^2} + 4x - 4x - {4^2}\]
The terms in between that is second and third term cancel each other as they have same coefficient but different signs or opposite signs
 \[ \Rightarrow {x^2} - {4^2}\]
This is of the form \[{a^2} - {b^2}\] . but as we discussed we will take the square of the numerical term if it occurs.
 \[ \Rightarrow {x^2} - 16\]
Thus standard form is \[\left( {x - 4} \right)\left( {x + 4} \right)\] \[ \Rightarrow {x^2} - 16\]
So, the correct answer is “ \[ {x^2} - 16\] ”.

Note: This type of problem is from algebraic expressions. They have identities of squares, cubes etc. standard form is nothing but upto that step from where it cannot be solved further. In these types of problems, be careful about the signs and brackets. As in addition as multiplications of two signs it matters so much. Note the table given below.

signs	addition	multiplication
\[ + , + \]	\[ + \]	\[ + \]
\[ + , - \]	\[ + \] (first number is greater)	\[ - \]
\[ - , + \]	\[ - \] (negative number is greater)	\[ - \]
\[ - , - \]	\[ - \]	\[ + \]