Question

How do you factor \[{x^2} - 81\] ?

Answer 1

Hint: We can solve this using algebraic identities. We use the identity \[{a^2} - {b^2} = (a - b)(a + b)\] to solve the given problem. We can see that 81 is a perfect square. We can convert the given problem into \[{a^2} - {b^2}\] form, since the square of 9 is 81 we can write 81 as \[{9^2}\] .

Complete step-by-step answer:
Given, \[{x^2} - 81\]
We can rewrite it as \[ = {x^2} - {9^2}\]
That is it is in the form \[{a^2} - {b^2}\] , where \[a = x\] and \[b = 9\] .
We have the formula \[{a^2} - {b^2} = (a - b)(a + b)\] .
Then above becomes,
 \[ = (x - 9)(x + 9)\] . These are the factors of the \[{x^2} - 81\] .
                               OR
We can find the root of the polynomial by equating the obtained factors to zero. That is
 \[ \Rightarrow (x - 9)(x + 9) = 0\]
Using the zero product principle, that is of ab=0 then a=0 or b=0.using this we get,
 \[ \Rightarrow x - 9 = 0\] and \[x + 9 = 0\] .
 \[ \Rightarrow x = 9\] and \[x = - 9\] . This is the roots of the given polynomial.
If we put the ‘x’ value in the above given problem it satisfies. Meaning that \[{x^2} - 81 = 0\]
We know that on the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x- intercept.
So, the correct answer is “ (x - 9)(x + 9)”.

Note: Since the given equation is a polynomial. The highest exponent of the polynomial in a polynomial equation is called its degree. A polynomial equation has exactly as many roots as its degree. Here the degree is 2. Hence it is called a quadratic equation. (We know the quadratic equation is of the form \[a{x^2} + bx + c = 0\] , in our problem coefficient of ‘x’ is zero) Hence, we have two roots or two factors.