Question

X^3-8x^2+3x-24=0 nnnnn

Answer 1

Ответ:

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Answer 2

To solve the equation x^3 - 8x^2 + 3x - 24 = 0, we can use various methods such as factoring, the rational root theorem, or numerical methods. Let's try using the rational root theorem to find the possible rational roots of the equation.

The rational root theorem states that any rational root of the equation has the form p/q, where p is a factor of the constant term (-24 in this case), and q is a factor of the leading coefficient (1 in this case).

The factors of the constant term (-24) are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24. The factors of the leading coefficient (1) are ±1.

Therefore, the possible rational roots are: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.

We can now test each of these values by substituting them into the equation to check if they are roots.

By substituting x = 1, we get: (1)^3 - 8(1)^2 + 3(1) - 24 = 1 - 8 + 3 - 24 = -28, which is not zero.

By substituting x = -1, we get: (-1)^3 - 8(-1)^2 + 3(-1) - 24 = -1 - 8 - 3 - 24 = -36, which is not zero.

By substituting x = 2, we get: (2)^3 - 8(2)^2 + 3(2) - 24 = 8 - 32 + 6 - 24 = -42, which is not zero.

By substituting x = -2, we get: (-2)^3 - 8(-2)^2 + 3(-2) - 24 = -8 - 32 - 6 - 24 = -70, which is not zero.

By substituting x = 3, we get: (3)^3 - 8(3)^2 + 3(3) - 24 = 27 - 72 + 9 - 24 = -60, which is not zero.

By substituting x = -3, we get: (-3)^3 - 8(-3)^2 + 3(-3) - 24 = -27 - 72 - 9 - 24 = -132, which is not zero.

By substituting x = 4, we get: (4)^3 - 8(4)^2 + 3(4) - 24 = 64 - 128 + 12 - 24 = -76, which is not zero.

By substituting x = -4, we get: (-4)^3 - 8(-4)^2 + 3(-4) - 24 = -64 - 128 - 12 - 24 = -228, which is not zero.

By substituting x = 6, we get: (6)^3 - 8(6)^2 + 3(6) - 24 = 216 - 288 + 18 - 24 = -78, which is not zero.

By substituting x = -6, we get: (-6)^3 - 8(-6)^2 + 3(-6) - 24 = -216 - 288 - 18 - 24 = -546, which is not zero.

By substituting x = 8, we get: (8)^3 - 8(8)^2 + 3(8) - 24 = 512 - 512 + 24 - 24 = 0.

Therefore, x = 8 is a root of the equation x^3 - 8x^2 + 3x - 24 = 0.

Now that we have found one root, we can divide the equation by (x - 8) using polynomial long division or synthetic division to obtain a quadratic equation. The quadratic equation can then be solved using the quadratic formula or factoring to find the remaining roots.