Question

How do you factor \[3{x^3} - 300x\] ?

Answer 1

Hint:We need to find the factors. First we take ‘3x’ common and we will have a quadratic equation. We can solve this using algebraic identities. We use the identity \[{a^2} - {b^2} = (a - b)(a + b)\]. Here we will have three factors or three roots (zeros). Because the degree of the equation is 3.

Complete step by step answer:
Given, \[3{x^3} - 300x\]
Taking common ‘3x’ we have,
\[3{x^3} - 300x = 3x({x^2} - 100)\]
We know that 100 is a perfect square and we can rewrite it as,
\[3x({x^2} - {10^2})\]
Since we know the algebraic identity \[{a^2} - {b^2} = (a - b)(a + b)\] and comparing with above we have \[a = x\] and \[b = 10\]. Then we have,
\[3x(x + 10)(x - 10)\]

Thus the factors of \[3{x^3} - 300x\] are \[(x + 10),(x - 10)\] and \[3x\].

Additional information:
We can also find the roots of the given equation. We substitute the above obtained factors equal to zero.
\[3x(x + 10)(x - 10) = 0\]
Using zero product principle we have,
\[ \Rightarrow 3x = 0\],\[x + 10 = 0\] and \[x - 10 = 0\].
\[ \Rightarrow x = 0\], \[x = - 10\] and \[x = 10\].
If we put the obtained answer in the above problem it satisfies the equation.

Note: Follow the same procedure for these kinds of problems. 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 3. Hence it is called a cubic equation. The general form of the cubic equation is \[a{x^3} + b{x^2} + cx + d = 0\]. The traditional way of solving a cubic equation is to reduce it to a quadratic equation and then we solve either by factoring or quadratic formula. That is what we did in the above given problem.