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How do you find the roots of ${x^3} - {x^2} - 17x + 15 = 0$?

Last updated date: 21st Jul 2024
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Hint: First we will reduce the equation further if possible. Then we will try to factorise the terms in the equation. Then solve the equation by using the expansion of ${a^2} - {b^2}$ and find the appropriate solution for the equation. Split the middle term and finally factorise the final term.

Complete step-by-step solution:
We will start off by reducing any reducible terms in the equation. But as you can see here, the equation ${x^3} - {x^2} - 17x + 15 = 0$ is not a perfect cube.
Now we know that the sum of the coefficients is $0$. That is
$1 + 1 - 17 + 15 = 0$
So, now we also know that, $x = 1$ is a root and $(x - 1)$ a factor. Now we will convert the cubic equation to a quadratic equation by splitting the middle term and then taking all the like terms together form two expressions.
  {x^3} - {x^2} - 17x + 15 = 0 \\
  (x - 1)({x^2} + 2x - 15) = 0 \\
Also, here note that $5 \times 3 = 15$ and $5 - 3 = 2$ so that,
   = {x^3} - {x^2} - 17x + 15 \\
   = (x - 1)(x + 5)(x - 3) \\
So, the other two roots are $x = - 5$ and $x = 3$.

Additional information: Rational roots test is one of the methods to solve such types of questions. It would only find Rational roots that are numbers $x$ which can be expressed as the quotient of two integers. The rational root theorem states that if a polynomial zero for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient.

Note: While splitting the middle term be careful. After splitting the middle term Do not solve all the equations simultaneously. Solve all the equations separately, so that you don’t miss any term of the solution. Check if the solution satisfies the original equation completely. If any term of the solution doesn’t satisfy the equation, then that term will not be considered as a part of the solution.