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<div class="item2">
    <h2>3. Riešenia úloh</h2>


    <section>



        <h3>Úlohy z kapitoly 3.2</h3>

        <ol>
            <li>
                <p>
                    -
                </p>
            </li>

            <li>
                <ol type="a">
                    <li>
                        <p>
                            Z dôvodu delenia nulou sa výpočet zrúti.
                        </p>
                    </li>

                    <li>
                        <p>
                            Stačí vložiť $\bf{B4 = 9}$.
                        </p>
                    </li>

                    <li>
                        <p>
                            Stačí vložiť $\bf{D6 = 12}$.
                        </p>
                    </li>

                    <li>
                        <p>

                            <details class="detail">
                                <summary class="summary"><b>Pozrite si 1. hárok zošitu MATICE</b></summary><br>
                                <div class="center2">
                                    <iframe scrolling="no" title="A3.04a" src="https://www.geogebra.org/material/iframe/id/d2pudvaq/width/944/height/916/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="944px" height="916px" style="border:0px;"> </iframe>
                                </div></details>
                        </p>
                    </li>
                </ol>

            </li>

            <li>
                <p>

                    <details class="detail">
                        <summary class="summary"><b>Pozrite si MATICE 03.</b></summary><br>
                        <div class="center2">
                            <iframe scrolling="no" title="A3.07" src="https://www.geogebra.org/material/iframe/id/mqesezwx/width/944/height/916/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="944px" height="916px" style="border:0px;"> </iframe>
                        </div></details>
                </p>
            </li>

            <li>
                <p>
                    Riešenie oboch sústav rovníc pre $\bf{n \in  {3, 4, 5}}$ nájdete v 1., resp.
                    v 2. hárku zošitu  Matice 04. Doriešte si sami minimálne prípad $\bf{n = 7}$,
                    ten je zaujímavý.
                    <details class="detail">
                        <summary class="summary"><b>1. hárok zošitu  Matice 04.</b></summary><br>
                        <div class="center2">
                            <iframe scrolling="no" title="A3.08a" src="https://www.geogebra.org/material/iframe/id/gejpyc7z/width/941/height/916/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="941px" height="916px" style="border:0px;"> </iframe>
                        </div></details>
                    <details class="detail">
                        <summary class="summary"><b>2. hárok zošitu  Matice 04.</b></summary><br>
                        <div class="center2">
                            <iframe scrolling="no" title="A3.08b" src="https://www.geogebra.org/material/iframe/id/jqmamp3q/width/941/height/916/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="941px" height="916px" style="border:0px;"> </iframe>
                        </div></details>
                </p>
            </li>

            <li>
                <p>
                    Hľadáme funkciu $\bf{f(x) = a|x+1| + b|x-1| + cx + d}$.
                </p>

                <p>
                    Vieme, že $\bf{f(-2) = 01, \; f(-1) = 1, \; f(1) = -1, \; f(2)=1}$.
                    Z toho dostávame sústavu rovníc:
                    \begin{array}{rr}
                    a & + & 3b & - & 2c & + & d & = & -1 \\
                    && 2b & - & c & + & d & = & 1 \\
                    2a & && + & c & + & d & = & -1 \\
                    3a & + & b & + & 2c & + & d & = & 1
                    \end{array}
                </p>

                <p>
                    Pomocou 2. listu zošitu MATICE 01 (keď tam dosadíte koeficienty sústavy) dostanete riešenie:
                    $$\bf{a = -1.5, \; b = 1.5, \; c = 2, \; d = 0}$$
                    <details class="detail">
                        <summary class="summary"><b>2. list zošitu MATICE 01</b></summary><br>
                        <div class="center2">
                            <iframe scrolling="no" title="A3.04b" src="https://www.geogebra.org/material/iframe/id/vzmjyuhn/width/944/height/916/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="944px" height="916px" style="border:0px;"> </iframe>
                        </div></details>
                </p>
            </li>

            <li>
                <p>
                    $$\bf{\frac{10x^2 - 16}{x^4-5x^2+4} = \frac{A}{x-2} + \frac{B}{x-1} + \frac{C}{x+1} + \frac{D}{x+2}}$$
                </p>

                <p>
                    Po sčítaní dostaneme zlomok, ktorého čitateľ je $\bf{K_3x^3 + K_2x^2 + K_1x + K_0}$,
                    pričom platí:
                    \begin{align}
                    &\bf{K_3 = A+B+C+D}\\
                    &\bf{K_2 = -[A(b+c+d) + B(a+c+d) + C(a+b+d) + D(a+b+c)]}\\
                    &\bf{K_1 = A(bc+cd+db) + B(ac+cd+da) + C(ab+bd+da) + D(ab+bc+ca)}\\
                    &\bf{K_0 = -(Abcd+aBcd+abCd+abcD)}
                    \end{align}
                </p>

                <p>
                    Ak dosadíme naše konkrétne hodnoty dostávame sústavu lineárnych rovníc:
                    \begin{align}
                    &\bf{K_3 = A+B+C+D = 0}\\
                    &\bf{K_2 = 2A + B - C - 2D = 10}\\
                    &\bf{K_1 = -A -4B - 4C - 1D = 0}\\
                    &\bf{K_0 = -2A - 4B + 4C + 2D = -16}\\
                    \end{align}
                </p>

                <p>
                    Pomocou 2. listu zošitu MATICE 01 sme po dosadení koeficientov sústavy dostali riešenie:
                    $$\bf{A=2, \; B=1,\;C=-1,\;D=-2}$$
                </p>
            </li>

            <li>
                <p>
                    Podľa úvah z 1. príkladu je $\bf{S_n^3 = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0}$.
                    Dostali sme sústavu rovníc (vľavo), ktorú sme riešili v zošite $S_n^3$.
                    <details class="detail">
                        <summary class="summary"><b>$S_n^3$</b></summary><br>
                        <div class="center2">
                            <iframe scrolling="no" title="A3.19" src="https://www.geogebra.org/material/iframe/id/tyeyxszx/width/944/height/916/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="944px" height="916px" style="border:0px;"> </iframe>
                        </div></details>
                    Po úpravách sme dostali(vpravo):
                    \begin{array}{cc}{
                    \begin{array}{:c:c:}
                    \hdashline\bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} \\\hdashline
                    \bf{16} & \bf{8} & \bf{4} & \bf{2} & \bf{1} & \bf{9} \\\hdashline
                    \bf{81} & \bf{27} & \bf{9} & \bf{3} & \bf{1} & \bf{36} \\\hdashline
                    \bf{256} & \bf{64} & \bf{16} & \bf{4} & \bf{1} & \bf{100} \\\hdashline
                    \bf{625} & \bf{125} & \bf{25} & \bf{5} & \bf{1} & \bf{225} \\\hdashline
                    \end{array}} &&

                    {\begin{array}{:c:c:}
                    \hdashline\bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} \\\hdashline
                    \bf{15} & \bf{7} & \bf{3} & \bf{1} & \bf{0} & \bf{8} \\\hdashline
                    \bf{50} & \bf{12} & \bf{2} & \bf{0} & \bf{0} & \bf{19} \\\hdashline
                    \bf{60} & \bf{6} & \bf{0} & \bf{0} & \bf{0} & \bf{18} \\\hdashline
                    \bf{24} & \bf{0} & \bf{0} & \bf{0} & \bf{0} & \bf{6} \\\hdashline
                    \end{array}}
                    \end{array}
                </p>

                <p>
                    Odtiaľ postupným dosadzovaním dostávame $\;\bf{a_4 = 1/4, \; a_3 = 2/4,\; a_2 = 1/4,\; a_1=0,\;a_0 =0}$.
                </p>
            </li>
        </ol>


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