# 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 |

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