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Find all real solutions of the polynomial equation:

[tex]x^4 + 10x^3 + 24x^2 - 19x - 70 = 0[/tex]

[tex]x = \square[/tex]

Answer :

We wish to solve the equation
[tex]$$
x^4 + 10x^3 + 24x^2 - 19x - 70 = 0.
$$[/tex]

A good approach is to factor the polynomial. It turns out that the polynomial factors as
[tex]$$
(x + 2)(x + 5)(x^2 + 3x - 7) = 0.
$$[/tex]

Since the product is zero, at least one of the factors must be zero. We solve each factor separately.

1. For the first factor:
[tex]$$
x + 2 = 0 \quad \Longrightarrow \quad x = -2.
$$[/tex]

2. For the second factor:
[tex]$$
x + 5 = 0 \quad \Longrightarrow \quad x = -5.
$$[/tex]

3. For the quadratic factor, we have:
[tex]$$
x^2 + 3x - 7 = 0.
$$[/tex]
We use the quadratic formula, which for an equation [tex]$ax^2+bx+c=0$[/tex] is given by:
[tex]$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$[/tex]
Here, [tex]$a=1$[/tex], [tex]$b=3$[/tex], and [tex]$c=-7$[/tex]. Substitute these values:
[tex]$$
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-7)}}{2(1)}
= \frac{-3 \pm \sqrt{9 + 28}}{2}
= \frac{-3 \pm \sqrt{37}}{2}.
$$[/tex]

Thus, the solutions to the polynomial equation are:
[tex]$$
x = -2,\quad x = -5,\quad x = \frac{-3 + \sqrt{37}}{2},\quad \text{and} \quad x = \frac{-3 - \sqrt{37}}{2}.
$$[/tex]

These are all the real solutions of the given equation.

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