Question

How do you solve \[{x^4} - 18{x^2} + 81 = 0\]?

Answer 1

Hint: 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. The roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts. If the polynomial of degree is ‘n’ then the number of roots or factors is ‘n’. We solve this using expanding the middle term.

Complete step-by-step answer:
The degree of the equation \[{x^4} - 18{x^2} + 81 = 0\]is 4, so the number of roots of the given equation is 4.
We can split the middle term ‘-18’ as ‘-9’ and ‘-9’. (The multiplication of -9 and -9 is +81. If we add -9 and -9 we get -18)
We get,
 \[{x^4} - 9{x^2} - 9{x^2} + 81 = 0\]
Now taking \[{x^2}\] common in the first two terms and taking -9 common in the last two terms we have,
 \[{x^2}({x^2} - 9) - 9({x^2} - 9) = 0\]
Again taking \[({x^2} - 9)\] common we get,
 \[({x^2} - 9)({x^2} - 9) = 0\]
This can rewrite it as
 \[({x^2} - {3^2})({x^2} - {3^2}) = 0\]
Now using the algebraic identity \[{a^2} - {b^2} = (a + b)(a - b)\] we get,
 \[ \Rightarrow (x + 3)(x - 3)(x + 3)(x - 3) = 0\]
Thus we have the four factors. Those are \[(x + 3)\], \[(x - 3)\], \[(x + 3)\] and \[(x - 3)\].
So, the correct answer is “ \[(x + 3)\], \[(x - 3)\], \[(x + 3)\] and \[(x - 3)\]”.

Note: We can also find the roots of the given polynomial. That is by equating each factor to zero. We get \[x = 3, - 3,3, - 3\](Four roots because the degree of the polynomial is 4). In various fields of mathematics require the point at which the value of a polynomial is zero, those values are called the factors/solution/zeros of the given polynomial. 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-intercepts. Careful in the calculation part.