Find all real solutions to the equation. x^3-5x^2-10x+50=0 Answer Altemptzout do (c) Additional Solution x=boxed (pm 10), x=square

To find all real solutions to the equation $$x^{3}-5x^{2}-10x+50=0$$ , we can use the Rational Root Theorem to identify possible rational roots and then use synthetic division or polynomial division to factor the polynomial.

Step 1: Use the Rational Root Theorem to identify possible rational roots.
The Rational Root Theorem states that if a polynomial has rational roots, then they must be factors of the constant term divided by factors of the leading coefficient. In this case, the constant term is 50 and the leading coefficient is 1, so the possible rational roots are $$\pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50$$ .

Step 2: Test the possible rational roots by substituting them into the polynomial.
Let's start with $$x = 1$$ :
$$1^{3} - 5(1)^{2} - 10(1) + 50 = 1 - 5 - 10 + 50 = 36 \neq 0$$
So, $$x = 1$$ is not a root.

Next, let's try $$x = -1$$ :
$$(-1)^{3} - 5(-1)^{2} - 10(-1) + 50 = -1 - 5 + 10 + 50 = 54 \neq 0$$
So, $$x = -1$$ is not a root.

Next, let's try $$x = 2$$ :
$$2^{3} - 5(2)^{2} - 10(2) + 50 = 8 - 20 - 20 + 50 = 18 \neq 0$$
So, $$x = 2$$ is not a root.

Next, let's try $$x = -2$$ :
$$(-2)^{3} - 5(-2)^{2} - 10(-2) + 50 = -8 - 20 + 20 + 50 = 42 \neq 0$$
So, $$x = -2$$ is not a root.

Next, let's try $$x = 5$$ :
$$5^{3} - 5(5)^{2} - 10(5) + 50 = 125 - 125 - 50 + 50 = 0$$
So, $$x = 5$$ is a root.

Step 3: Use synthetic division to factor the polynomial.
We can now use synthetic division to divide the polynomial by $$(x - 5)$$ and find the other factors.

$$\begin{array}{r|rrrr} 5 & 1 & -5 & -10 & 50 \\ & & 5 & 0 & -50 \\ \hline & 1 & 0 & -10 & 0 \\ \end{array}$$

The quotient is $$x^{2} - 10$$ , so we can write the polynomial as:
$$x^{3} - 5x^{2} - 10x + 50 = (x - 5)(x^{2} - 10)$$

Step 4: Solve for the remaining roots.
To find the remaining roots, we set $$x^{2} - 10 = 0$$ and solve for $$x$$ :
$$x^{2} - 10 = 0$$
$$x^{2} = 10$$
$$x = \pm \sqrt{10}$$

Therefore, the real solutions to the equation $$x^{3} - 5x^{2} - 10x + 50 = 0$$ are:
$$x = 5, x = \sqrt{10}, x = -\sqrt{10}$$