Polynomial equation solver

This calculator solves equations that are reducible to polynomial form , such as $ \color{blue}{2(x+1) + 3(x-1) = 5} $ , $ \color{blue}{(2x+1)^2 - (x-1)^2 = x} $ and $ \color{blue}{ \frac{2x+1}{2} + \frac{3-4x}{3} = 1} $ . The calculator will try to find an exact solution; if this is not possible, it will use the cubic or quartic formulas. The calculator will walk you through each step and give you a detailed explanation on how to simplify and solve the equation.

Solve equation$$-x^3-x^2+21x+45 = 0$$

solution

$$ \begin{matrix}x_1 = -3 & x_2 = 5 \\[1 em] \end{matrix} $$

explanation

$ \color{blue}{ -x^{3}-x^{2}+21x+45 } $ is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $ \dfrac{p}{q} $, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factor of the leading coefficient ( -1 ) is 1 .The factors of the constant term (45) are 1 3 5 9 15 45 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 45 }{ 1 } ~ $$

Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial $ P(x) $, we obtain $ P(-3) = 0 $.

To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $ \color{blue}{q x - p} $. In this example:

Divide $ P(x) $ with $ \color{blue}{x + 3} $

$$ \frac{ -x^{3}-x^{2}+21x+45 }{ \color{blue}{ x + 3 } } = -x^{2}+2x+15 $$

Polynomial $ -x^{2}+2x+15 $ can be used to find the remaining roots.

$ \color{blue}{ -x^{2}+2x+15 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.