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# System prepended metadata

title: Wielomiany

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# Wielomiany

a) $4x^3 - 4x^2 - 15x + 18 = 0$



| --  | 4    | -4   | -15 | 18  |
| --- | ---- | ---- | --- | --- |
| -2  | 4    | -12  | 9   | 0   |

$(x+2)(4x^2 - 12x + 9) = 0$

**I SPOSÓB – WZÓR SKRÓCONEGO MNOŻENIA**
$(a-b)^2 = a^2 - 2ab + b^2$

$4x^2 - 12x + 9 = 0$
$(2x -3)^2 = 0$

$2x - 3 = 0$
$2x = 3$
$x = \frac{3}{2}$

**II SPOSÓB – DELTA**
$4x^2 - 12x + 9 = 0$

$\Delta = 144 - 4 * 4 * 9 = 0$
$x = \frac{3}{2}$

b) $-3x^3 + 8x^2 + x = 0$

$x(-3x^2 + 8x + 1) = 0$

$\Delta = 64 + 12 = 76$

c) $x^3 + x^2 - 2x - 2 = 0$

$x^2(x + 1) - 2(x + 1) = 0$
$(x+1)(x^2 - 2) = 0$
$(x+1)(x - \sqrt{2})(x + \sqrt{2}) = 0$

$x = -1$
$x = - \sqrt{2}$
$x = \sqrt{2}$

d) $3x + 5x^4 - 3x^5 = 5$

$3x + 5x^4 - 3x^5 - 5 = 0$
$5x^4 - 5 - 3x^5 + 3x = 0$
$5(x^4 - 1) - 3x(x^4 - 1) = 0$
$(x^4 - 1)(5 - 3x) = 0$

$x^4 - 1 = 0$ lub $5 - 3x = 0$
$x^4 = 1$
$x = 1$ lub $x = -1$ lub $x = \frac{5}{3}$

$a^2 - b^2 = (a-b)(a+b)$
$(x^4 - 1) = (x^2 -1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1)$

$(x - 1)(x + 1)(x^2 + 1)(5 - 3x) = 0$
$-(3x - 5)(x - 1)(x + 1)(x^2 + 1) = 0$


PRZYKŁAD INNY
$2x^3 - x^2 - 8x + 4 = 0$
dzielimy przez 2 schematem Hornera


| --- | 2 | -1  | -8  | 4   |
| --- | - | --- | --- | --- |
| 2   | 2 | 3   | -2  | 0   |

$(2x^2 + 3x - 2)(x-2) = 0$
$\Delta = 9 + 16 = 25$
$\sqrt{\Delta} = \sqrt{25} = 5$
