https://peda.net/id/7cbc9c8cbbd01 2026-09-29T09:39:14+03:00 https://peda.net/:static/548/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license noopener" href="https://peda.net/info" target="_blank">Peda.net-yleislisenssi</a></div> https://peda.net/id/7cbd9f76bbd01 2026-09-29T09:40:33+03:00 <div> </div> <div>Polynomilaskennassa muistikaavoilla tarkoitetaan binomin neliötä sekä summan ja erotuksen tuloa. Binomi on polynomi, jossa on kaksi termiä (yhteenlaskettavaa)</div> <div> </div> <div><span class="editor underline">1. Binomin neliö</span></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a%2Bb%5Cright)%5E2%3Da%5E2%2B2ab%2Bb%5E2" alt="\left(a+b\right)^2=a^2+2ab+b^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cleft(a%2Bb%5Cright)%5E2%3D%5Cleft(a%2Bb%5Cright)%5Cleft(a%2Bb%5Cright)%3Da%5E2%2Bab%2Bba%2Bb%5E2%3Da%5E2%2B2ab%2Bb%5E2%5Cright)" alt="\left(\left(a+b\right)^2=\left(a+b\right)\left(a+b\right)=a^2+ab+ba+b^2=a^2+2ab+b^2\right)"/><br/> <div>Huom! </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5E2%5C%20ja%5C%20b%5E2%5C%20ovat%5C%20aina%5C%20etumerkilt%C3%A4%C3%A4n%5C%20positiivisia" alt="a^2\ ja\ b^2\ ovat\ aina\ etumerkiltään\ positiivisia"/></div> <span>termien a ja b kaksinkertainen tulo<br/> </span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2ab%5C%20on%5C%20etumerkilt%C3%A4%C3%A4n%5C%20positiivinen%7B%2C%7D%5C%20jos%5C%20molemmat%5C%20termit%5C%20ovat%5C%20samanmerkkisi%C3%A4%5C%20ja" alt="2ab\ on\ etumerkiltään\ positiivinen{,}\ jos\ molemmat\ termit\ ovat\ samanmerkkisiä\ ja"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2ab%5C%20%3C0%7B%2C%7D%5C%20kun%5C%20a%5C%20ja%5C%20b%5C%20ovat%5C%20erimerkkiset" alt="2ab\ &lt;0{,}\ kun\ a\ ja\ b\ ovat\ erimerkkiset"/></div> <div> </div> <span>Sievennä</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B3%5Cright)%5E2%3Dx%5E2%2B2%5Ccdot3x%2B3%5E2%3Dx%5E2%2B6x%2B9" alt="\left(x+3\right)^2=x^2+2\cdot3x+3^2=x^2+6x+9"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x-5%5Cright)%5E2%3D%5Cleft(2x%5Cright)%5E2%2B2%5Ccdot%5Cleft(-10x%5Cright)%2B%5Cleft(-5%5Cright)%5E2%3D4x%5E2-20x%2B25" alt="\left(2x-5\right)^2=\left(2x\right)^2+2\cdot\left(-10x\right)+\left(-5\right)^2=4x^2-20x+25"/><br/> <br/> </div> <div> </div> <div><span class="editor underline">2. Summan ja erotuksen tulo</span></div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%3Da%5E2-b%5E2" alt="\left(a+b\right)\left(a-b\right)=a^2-b^2"/></div> <div>esimerkki</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-4%5Cright)%5Cleft(x%2B4%5Cright)%3Dx%5E2-4%5E2%3Dx%5E2-16" alt="\left(x-4\right)\left(x+4\right)=x^2-4^2=x^2-16"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x-2%5Cright)%5Cleft(3x%2B2%5Cright)%3D9x%5E2-4" alt="\left(3x-2\right)\left(3x+2\right)=9x^2-4"/></div> <div><br/> <br/> </div> 2026-09-29T09:39:14+03:00 https://peda.net/id/7cbe0c73bbd01 2026-09-29T09:41:27+03:00 <span>Laske/sievennä</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2-3x%2B5-2%5Cleft(-x%5E2%2B3x-1%5Cright)%3D2x%5E2-3x%2B5%2B2x%5E2-6x%2B2%3D" alt="2x^2-3x+5-2\left(-x^2+3x-1\right)=2x^2-3x+5+2x^2-6x+2="/></div> <span>(yhdistetään samanmuotoiset eli samanasteiset termit eli</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%5E2-%7B%2C%7D%5C%20x-%5C%20ja%5C%20vakiotermit%5Cright)" alt="\left(x^2-{,}\ x-\ ja\ vakiotermit\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D4x%5E2-9x%2B7" alt="=4x^2-9x+7"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5Cleft(4x%5E2-x%2B6%5Cright)%3D3%5Ccdot4%5Ccdot%20x%5Ccdot%20x%5E2-3x%5Ccdot%20x%2B6%5Ccdot3x%3D12x%5E3-3x%5E2%2B18x" alt="3x\left(4x^2-x+6\right)=3\cdot4\cdot x\cdot x^2-3x\cdot x+6\cdot3x=12x^3-3x^2+18x"/></div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a-3%5Cright)%5Cleft(2a%2B1%5Cright)%3Da%5Ccdot2a%2Ba%5Ccdot1-3%5Ccdot2a-3%5Ccdot1%3D2a%5E2%2Ba-6a-3" alt="\left(a-3\right)\left(2a+1\right)=a\cdot2a+a\cdot1-3\cdot2a-3\cdot1=2a^2+a-6a-3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2a%5E2-5a-3" alt="=2a^2-5a-3"/><br/> <h3><br/> <br/> <span>Sievennä</span></h3> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(x%2B3%5Cright)-%5Cleft(x-3%5Cright)%5Cleft(x%2B2%5Cright)%3Dx%5E2%2B3x-%5Cleft(x%5E2%2B2x-3x-6%5Cright)" alt="x\left(x+3\right)-\left(x-3\right)\left(x+2\right)=x^2+3x-\left(x^2+2x-3x-6\right)"/></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=%3Dx%5E2%2B3x-x%5E2-2x%2B3x%2B6%3D4x%2B6" alt="=x^2+3x-x^2-2x+3x+6=4x+6"/></h3> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(x%2B1%5Cright)%5Cleft(x-2%5Cright)%3Dx%5Cleft(x%5E2-2x%2Bx-2%5Cright)%3Dx%5Cleft(x%5E2-x-2%5Cright)%3Dx%5E3-x%5E2-2x" alt="x\left(x+1\right)\left(x-2\right)=x\left(x^2-2x+x-2\right)=x\left(x^2-x-2\right)=x^3-x^2-2x"/></div> <h3><span>c) Laske seuraavan lausekkeen arvo, kun a = -3</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2a-2%5Cright)%5Cleft(2%2B2a%5Cright)-%5Cleft(2a%5E2%2B3a-5%5Cright)%2B3a%5Cleft(a%2B1%5Cright)" alt="\left(2a-2\right)\left(2+2a\right)-\left(2a^2+3a-5\right)+3a\left(a+1\right)"/></div> <h3><span>Sievennetään ensin lauseketta avaamalla sulkeet ja yhdistämällä samanmuotoiset termit</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D4a%2B4a%5E2-4-4a-2a%5E2-3a%2B5%2B3a%5E2%2B3a%3D5a%5E2%2B1" alt="=4a+4a^2-4-4a-2a^2-3a+5+3a^2+3a=5a^2+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=nyt%5C%20kun%5C%20t%C3%A4h%C3%A4n%5C%20sijoitetaan%5C%20a%5C%20%3D%5C%20-3%5C%20saadaan" alt="nyt\ kun\ tähän\ sijoitetaan\ a\ =\ -3\ saadaan"/></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ccdot%5Cleft(-3%5Cright)%5E2%2B1%3D46" alt="5\cdot\left(-3\right)^2+1=46"/></h3> <div> </div> <div>Jaa polynomi tekijöihin</div> <div>Etsitään polynomista yhteistä tekijää, jonka avulla polynomi voidaan kirjoittaa tulomuotoon</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-4x%3Dx%5Ccdot%20x-4%5Ccdot%20x%3Dx%5Cleft(x-4%5Cright)" alt="x^2-4x=x\cdot x-4\cdot x=x\left(x-4\right)"/></div> <h3><span>b)</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x-9x%5E2%3D3x%5Cleft(1-3x%5Cright)" alt="3x-9x^2=3x\left(1-3x\right)"/></div> <h3><span>c)</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B1%5Cright)%5Cleft(x-4%5Cright)%2Bx%5Cleft(x-4%5Cright)" alt="\left(x+1\right)\left(x-4\right)+x\left(x-4\right)"/></div> <h3><span>Huomataan että molemmilla on nyt yhteisenä tekijänä x - 4</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(x-4%5Cright)%5Cleft(x%2B1%2Bx%5Cright)%3D%5Cleft(x-4%5Cright)%5Cleft(2x%2B1%5Cright)" alt="=\left(x-4\right)\left(x+1+x\right)=\left(x-4\right)\left(2x+1\right)"/></div> <div> </div> <div> </div> <div>Sievennä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-4%5Cright)%5Cleft(x%2B4%5Cright)-%5Cleft(x%2B4%5Cright)%5E2%3Dx%5E2%2B4x-4x-16-%5Cleft(x%2B4%5Cright)%5Cleft(x%2B4%5Cright)" alt="\left(x-4\right)\left(x+4\right)-\left(x+4\right)^2=x^2+4x-4x-16-\left(x+4\right)\left(x+4\right)"/></div> <h3><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3Dx%5E2-16-%5Cleft(x%5E2%2B4x%2B4x%2B16%5Cright)%3Dx%5E2-16-x%5E2-8x-16" alt="=x^2-16-\left(x^2+4x+4x+16\right)=x^2-16-x^2-8x-16"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-8x-32" alt="=-8x-32"/><br/> <br/> <br/> <br/> <b>Muistikaavat</b><br/> <br/> </h3> <div>Polynomilaskennassa muistikaavoilla tarkoitetaan binomin neliötä sekä summan ja erotuksen tuloa. Binomi on polynomi, jossa on kaksi termiä (yhteenlaskettavaa)</div> <div> </div> <div><span class="editor underline">1. Binomin neliö</span></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a%2Bb%5Cright)%5E2%3Da%5E2%2B2ab%2Bb%5E2" alt="\left(a+b\right)^2=a^2+2ab+b^2"/></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cleft(a%2Bb%5Cright)%5E2%3D%5Cleft(a%2Bb%5Cright)%5Cleft(a%2Bb%5Cright)%3Da%5E2%2Bab%2Bba%2Bb%5E2%3Da%5E2%2B2ab%2Bb%5E2%5Cright)" alt="\left(\left(a+b\right)^2=\left(a+b\right)\left(a+b\right)=a^2+ab+ba+b^2=a^2+2ab+b^2\right)"/></h3> <div>Huom! </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5E2%5C%20ja%5C%20b%5E2%5C%20ovat%5C%20aina%5C%20etumerkilt%C3%A4%C3%A4n%5C%20positiivisia" alt="a^2\ ja\ b^2\ ovat\ aina\ etumerkiltään\ positiivisia"/></div> <h3><span>termien a ja b kaksinkertainen tulo</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2ab%5C%20on%5C%20etumerkilt%C3%A4%C3%A4n%5C%20positiivinen%7B%2C%7D%5C%20jos%5C%20molemmat%5C%20termit%5C%20ovat%5C%20samanmerkkisi%C3%A4%5C%20ja" alt="2ab\ on\ etumerkiltään\ positiivinen{,}\ jos\ molemmat\ termit\ ovat\ samanmerkkisiä\ ja"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2ab%5C%20%3C0%7B%2C%7D%5C%20kun%5C%20a%5C%20ja%5C%20b%5C%20ovat%5C%20erimerkkiset" alt="2ab\ &lt;0{,}\ kun\ a\ ja\ b\ ovat\ erimerkkiset"/></div> <div><span class="editor underline">2. Summan ja erotuksen tulo</span></div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%3Da%5E2-b%5E2" alt="\left(a+b\right)\left(a-b\right)=a^2-b^2"/><br/> <br/> <br/> <span>Sievennä käyttäen muistikaavoja</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x-3%5Cright)%5E2%3D%5C%20%5C%20%5Cleft(kyseess%C3%A4%5C%20bino%5Cmin%5C%20neli%C3%B6%5C%20%5Cleft(a%2Bb%5Cright)%5E2%3Da%5E2%2B2ab%2Bb%5E2%5Cright)" alt="\left(2x-3\right)^2=\ \ \left(kyseessä\ bino\min\ neliö\ \left(a+b\right)^2=a^2+2ab+b^2\right)"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x%5Cright)%5E2%2B2%5Ccdot2x%5Ccdot%5Cleft(-3%5Cright)%2B%5Cleft(-3%5Cright)%5E2%3D4x%5E2-12x%2B9" alt="\left(2x\right)^2+2\cdot2x\cdot\left(-3\right)+\left(-3\right)^2=4x^2-12x+9"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x-4%5Cright)%5Cleft(3x%2B4%5Cright)%3D%5Cleft(3x%5Cright)%5E2-4%5E2%3D9x%5E2-16" alt="\left(3x-4\right)\left(3x+4\right)=\left(3x\right)^2-4^2=9x^2-16"/></div> <span>nyt kyseessä on summan ja erotuksen tulo</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a-b%5Cright)%5Cleft(a%2Bb%5Cright)%3Da%5E2-b%5E2" alt="\left(a-b\right)\left(a+b\right)=a^2-b^2"/></div> <div> </div> <div> </div> <div>Sievennä</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B3%5Cright)%5E2%2Bx%5Cleft(x%2B2%5Cright)-2%5Cleft(x-3%5Cright)%5E2%3Dx%5E2%2B2%5Ccdot3x%2B9%2Bx%5E2%2B2x-2%5Cleft(x%5E2-2%5Ccdot3x%2B9%5Cright)" alt="\left(x+3\right)^2+x\left(x+2\right)-2\left(x-3\right)^2=x^2+2\cdot3x+9+x^2+2x-2\left(x^2-2\cdot3x+9\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3Dx%5E2%2B6x%2B9%2Bx%5E2%2B2x-2x%5E2%2B12x-18" alt="=x^2+6x+9+x^2+2x-2x^2+12x-18"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D18x-9" alt="=18x-9"/><br/> <div> </div> </div> <h3><br/> <span>Sievennä</span></h3> <div>a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5C%20%5Cleft(5x%2B2%5Cright)%5Cleft(2-5x%5Cright)%2B%5Cleft(5x%2B2%5Cright)%5E2%3D%5Cleft(2%2B5x%5Cright)%5Cleft(2-5x%5Cright)%2B25x%5E2%2B2%5Ccdot5x%5Ccdot2%2B4" alt="\ \left(5x+2\right)\left(2-5x\right)+\left(5x+2\right)^2=\left(2+5x\right)\left(2-5x\right)+25x^2+2\cdot5x\cdot2+4"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D4-25x%5E2%2B25x%5E2%2B20x%2B4%3D20x%2B8" alt="=4-25x^2+25x^2+20x+4=20x+8"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%5Cleft(x%5E2%2B1%5Cright)%3D3%5Cleft(x%5E2-1%5Cright)%5Cleft(x%5E2%2B1%5Cright)" alt="3\left(x+1\right)\left(x-1\right)\left(x^2+1\right)=3\left(x^2-1\right)\left(x^2+1\right)"/></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=%3D3%5Cleft(x%5E4-1%5Cright)%3D3x%5E4-3" alt="=3\left(x^4-1\right)=3x^4-3"/></h3> <div> </div> <div>Ratkaise yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=19-%5Cleft(y%2B2%5Cright)%5E2%3D-2%5Cleft(y%2B1%5Cright)%5Cleft(y-1%5Cright)%2B%5Cleft(1%2By%5Cright)%5E2" alt="19-\left(y+2\right)^2=-2\left(y+1\right)\left(y-1\right)+\left(1+y\right)^2"/></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=19-%5Cleft(y%5E2%2B4y%2B4%5Cright)%3D-2%5Cleft(y%5E2-1%5Cright)%2B1%2B2y%2By%5E2" alt="19-\left(y^2+4y+4\right)=-2\left(y^2-1\right)+1+2y+y^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=19-y%5E2-4y-4%3D-2y%5E2%2B2%2B1%2B2y%2By%5E2" alt="19-y^2-4y-4=-2y^2+2+1+2y+y^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-6y%3D-12%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20y%3D2" alt="-6y=-12\ \ \Leftrightarrow\ \ y=2"/></h3> <div> </div> <h3><br/> <br/> </h3> <div><b>NELIÖJUURI</b></div> <div> </div> <div>Luvun a ≥ 0 (ei negatiivisen luvun a) neliöjuuri, merkitään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7Ba%7D" alt="\sqrt{a}"/></div> <h3><span>on luku, joka toteuttaa ehdot:</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1.%5C%20%5Csqrt%7Ba%7D%5Cge0%5C%20%5C%20ja%5C%20%5C%202.%5C%20%5Cleft(%5Csqrt%7Ba%7D%5Cright)%5E2%3Da" alt="1.\ \sqrt{a}\ge0\ \ ja\ \ 2.\ \left(\sqrt{a}\right)^2=a"/></div> <h3><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=siis%5C%20%5Csqrt%7Ba%7D%3Db%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20b%5Cge0%5C%20%5C%20ja%5C%20%5C%20b%5E2%3Da" alt="siis\ \sqrt{a}=b\ \ \Leftrightarrow\ \ b\ge0\ \ ja\ \ b^2=a"/></h3> <div>esimerkiksi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B25%7D%3D5%7B%2C%7D%5C%20sill%C3%A4%5C%205%5E2%3D25%5C%20%5C%20ja%5C%205%5C%20%3E0" alt="\sqrt{25}=5{,}\ sillä\ 5^2=25\ \ ja\ 5\ &gt;0"/><!--filtered attribute: style="max-width: 100%; border: none;"--></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B-9%7D%5C%20%5C%20ei%5C%20ole%5C%20m%C3%A4%C3%A4ritelty%7B%2C%7D%5C%20sill%C3%A4%5C%20-9%3C0" alt="\sqrt{-9}\ \ ei\ ole\ määritelty{,}\ sillä\ -9&lt;0"/></h3> <div> </div> <div>Ratkaise yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5x%5E2-80%3D0%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%205x%5E2%3D80%5C%20%5C%20%5Cmid%3A5" alt="5x^2-80=0\ \ \Leftrightarrow\ \ 5x^2=80\ \ \mid:5"/></div> <h3><img src="https://math-demo.abitti.fi/math.svg?latex=%5CLeftrightarrow%5C%20x%5E2%3D16%5C%20%5C%20%5Cparallel%5C%20%5Csqrt%7B%20%7D" alt="\Leftrightarrow\ x^2=16\ \ \parallel\ \sqrt{ }"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm%5Csqrt%7B16%7D%3D%5Cpm4%5C%20%5C%20%5C%20%5Cleft(x%3D4%5C%20tai%5C%20x%5C%20%3D-4%5Cright)" alt="x=\pm\sqrt{16}=\pm4\ \ \ \left(x=4\ tai\ x\ =-4\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=sill%C3%A4%5C%204%5E2%3D16%5C%20%5C%20ja%5C%20%5Cleft(-4%5Cright)%5E2%3D16" alt="sillä\ 4^2=16\ \ ja\ \left(-4\right)^2=16"/></h3> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Yht%C3%A4l%C3%B6n%5C%20%5C%20x%5E2%3Da%5C%20ratkaisut%7B%2C%7D%5C%20kun%5C%20a%5Cge0%5C%20ovat%5C%20%5C%20x%3D%5Cpm%5Csqrt%7Ba%7D" alt="Yhtälön\ \ x^2=a\ ratkaisut{,}\ kun\ a\ge0\ ovat\ \ x=\pm\sqrt{a}"/></div> <h3><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=Huom!%5C%20%5C%20Yht%C3%A4l%C3%B6ll%C3%A4%5C%20x%5E2%3D-8%5C%20ei%5C%20ole%5C%20ratkaisuja%7B%2C%7D%5C%20sill%C3%A4" alt="Huom!\ \ Yhtälöllä\ x^2=-8\ ei\ ole\ ratkaisuja{,}\ sillä"/></h3> <div>negatiivisesta luvusta ei voida ottaa neliöjuurta eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B-8%7D%5C%20%5C%20ei%5C%20ole%5C%20m%C3%A4%C3%A4ritelty" alt="\sqrt{-8}\ \ ei\ ole\ määritelty"/></div> <h3><span>(voidaan myös perustella siten, että koska neliö</span></h3> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2on%5C%20%5C%20aina%5C%20positiivinen%5C%20niin%5C%20x%5E2%3D-8%5C%20on%5C%20mahdoton" alt="x^2on\ \ aina\ positiivinen\ niin\ x^2=-8\ on\ mahdoton"/></div> <div> </div> <div>Neliöjuuren laskusääntöjä</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1.%5C%20%5Cleft(%5Csqrt%7Ba%7D%5Cright)%5E2%3Da%5C%20%5C%20%5C%20ja%5C%20%5C%20%5Cleft(2.%5C%20%5Csqrt%7Ba%5E2%7D%3D%5Cleft%7Ca%5Cright%7C%5C%20%5Cleft(itseisarvo%5C%20a%3Asta%5Cright)%5Cright)" alt="1.\ \left(\sqrt{a}\right)^2=a\ \ \ ja\ \ \left(2.\ \sqrt{a^2}=\left|a\right|\ \left(itseisarvo\ a:sta\right)\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3.%5C%20%5Csqrt%7Bab%7D%3D%5Csqrt%7Ba%7D%5Csqrt%7Bb%7D%5C%20%5C%20%7B%2C%7D4.%5C%20%5C%20%5C%20%5Csqrt%7B%5Cfrac%7Ba%7D%7Bb%7D%7D%3D%5Cfrac%7B%5Csqrt%7Ba%7D%7D%7B%5Csqrt%7Bb%7D%7D" alt="3.\ \sqrt{ab}=\sqrt{a}\sqrt{b}\ \ {,}4.\ \ \ \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5.%5C%20%5Csqrt%7Ba%7D%3E%5Csqrt%7Bb%7D%7B%2C%7D%5C%20jos%5C%20a%3Eb" alt="5.\ \sqrt{a}&gt;\sqrt{b}{,}\ jos\ a&gt;b"/><!--filtered attribute: style="max-width: 100%; border: none;"--><br/> <br/> <br/> <span>Sievennä</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B18%7D%3D%5Csqrt%7B9%5Ccdot2%7D%3D%5Csqrt%7B9%7D%5Ccdot%5Csqrt%7B2%7D%3D3%5Csqrt%7B2%7D" alt="\sqrt{18}=\sqrt{9\cdot2}=\sqrt{9}\cdot\sqrt{2}=3\sqrt{2}"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B%5Cfrac%7B8%7D%7B25%7D%7D%3D%5Cfrac%7B%5Csqrt%7B8%7D%7D%7B%5Csqrt%7B25%7D%7D%3D%5Cfrac%7B%5Csqrt%7B4%5Ccdot2%7D%7D%7B5%7D%3D%5Cfrac%7B2%5Csqrt%7B2%7D%7D%7B5%7D%5Capprox" alt="\sqrt{\frac{8}{25}}=\frac{\sqrt{8}}{\sqrt{25}}=\frac{\sqrt{4\cdot2}}{5}=\frac{2\sqrt{2}}{5}\approx"/></div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csqrt%7B3%7D%5Ccdot4%5Csqrt%7B6%7D%3D2%5Ccdot4%5Ccdot%5Csqrt%7B3%5Ccdot6%7D%3D8%5Csqrt%7B9%5Ccdot2%7D%3D8%5Ccdot3%5Csqrt%7B2%7D%3D24%5Csqrt%7B2%7D" alt="2\sqrt{3}\cdot4\sqrt{6}=2\cdot4\cdot\sqrt{3\cdot6}=8\sqrt{9\cdot2}=8\cdot3\sqrt{2}=24\sqrt{2}"/></div> <span>d)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csqrt%7B5%7D-%5Csqrt%7B3%7D%2B4%5Csqrt%7B3%7D%2B%5Csqrt%7B5%7D%3D3%5Csqrt%7B5%7D%2B3%5Csqrt%7B3%7D%5Capprox" alt="2\sqrt{5}-\sqrt{3}+4\sqrt{3}+\sqrt{5}=3\sqrt{5}+3\sqrt{3}\approx"/></div> <br/> <div> </div> <br/> <span>Huom! Nimittäjässä oleva neliöjuuri pyritään murtoluvuissa &quot;laventamaan pois&quot;</span> <div>Poikkeuksena muotoa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B%5Csqrt%7Ba%7D%7D%5C%20ei%5C%20v%C3%A4ltt%C3%A4m%C3%A4tt%C3%A4%5C%20tarvi%5C%20laventaa%5C%20%5Cleft(sievent%C3%A4%C3%A4%5Cright)" alt="\frac{1}{\sqrt{a}}\ ei\ välttämättä\ tarvi\ laventaa\ \left(sieventää\right)"/></div> <span>esimerkiksi</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B6%7D%7B%5Csqrt%7B2%7D%7D%3D%5Cfrac%7B6%5Csqrt%7B2%7D%7D%7B%5Cleft(%5Csqrt%7B2%7D%5Cright)%5E2%7D%3D%5Cfrac%7B6%5Csqrt%7B2%7D%7D%7B2%7D%3D3%5Csqrt%7B2%7D" alt="\frac{6}{\sqrt{2}}=\frac{6\sqrt{2}}{\left(\sqrt{2}\right)^2}=\frac{6\sqrt{2}}{2}=3\sqrt{2}"/></div> <span><span>Kotitehtävät: 170 - 173<br/> <br/> </span></span><span><span><br/> </span></span> <div> </div> <div>Sievennä</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B10%7D%7B%5Csqrt%7B5%7D%7D%3D%5Cfrac%7B10%5Csqrt%7B5%7D%7D%7B%5Csqrt%7B5%7D%5E2%7D%3D%5Cfrac%7B10%5Csqrt%7B5%7D%7D%7B5%7D%3D2%5Csqrt%7B5%7D" alt="\frac{10}{\sqrt{5}}=\frac{10\sqrt{5}}{\sqrt{5}^2}=\frac{10\sqrt{5}}{5}=2\sqrt{5}"/></div> <span><span>b)</span></span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Csqrt%7B5%7D-1%5Cright)%5E2%2B2%5Csqrt%7B5%7D-%5Cleft(2-%5Csqrt%7B3%7D%5Cright)%5Cleft(2%2B%5Csqrt%7B3%7D%5Cright)%3D%5Csqrt%7B5%7D%5E2-2%5Csqrt%7B5%7D%2B1%2B2%5Csqrt%7B5%7D-%5Cleft(2%5E2-%5Csqrt%7B3%7D%5E2%5Cright)" alt="\left(\sqrt{5}-1\right)^2+2\sqrt{5}-\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)=\sqrt{5}^2-2\sqrt{5}+1+2\sqrt{5}-\left(2^2-\sqrt{3}^2\right)"/></div> <span><span><img src="https://math-demo.abitti.fi/math.svg?latex=%3D5%2B1-%5Cleft(4-3%5Cright)%3D5" alt="=5+1-\left(4-3\right)=5"/><br/> </span></span> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%7D%7B3%2B%5Csqrt%7B6%7D%7D%3D" alt="\frac{3}{3+\sqrt{6}}="/></div> <span><span>Nimittäjässä oleva neliöjuuri &quot;poistetaan&quot; laventamalla</span></span> <div>(kokeillaan, jos lavennetaan </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%2B%5Csqrt%7B6%7D" alt="3+\sqrt{6}"/></div> <span><img src="https://math-demo.abitti.fi/math.svg?latex=nyt%5C%20nimitt%C3%A4j%C3%A4%C3%A4n%5C%20tulisi%5C%20%5Cleft(3%2B%5Csqrt%7B6%7D%5Cright)%5E2%3D3%5E2%2B6%5Csqrt%7B6%7D%2B%5Csqrt%7B6%7D%5E2%5C%20%5CRightarrow%5C%20eli%5C%20neli%C3%B6juuri%5C%20ei%5C%20%22h%C3%A4vi%C3%A4%22" alt="nyt\ nimittäjään\ tulisi\ \left(3+\sqrt{6}\right)^2=3^2+6\sqrt{6}+\sqrt{6}^2\ \Rightarrow\ eli\ neliöjuuri\ ei\ &quot;häviä&quot;"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=Ainoa%5C%20tapa%5C%20on%5C%20laventaa%5C%20siten%7B%2C%7D%5C%20ett%C3%A4%5C%20tulee%5C%20summan%5C%20ja%5C%20erotuksen%5C%20tulo%5C%20%5Cleft(a-b%5Cright)%5Cleft(a%2Bb%5Cright)%3Da%5E2-b%5E2" alt="Ainoa\ tapa\ on\ laventaa\ siten{,}\ että\ tulee\ summan\ ja\ erotuksen\ tulo\ \left(a-b\right)\left(a+b\right)=a^2-b^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=eli%5C%20lavennetaan%5C%203-%5Csqrt%7B6%7D" alt="eli\ lavennetaan\ 3-\sqrt{6}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%7D%7B3%2B%5Csqrt%7B6%7D%7D%3D%5Cfrac%7B3%5Cleft(3-%5Csqrt%7B6%7D%5Cright)%7D%7B%5Cleft(3%2B%5Csqrt%7B6%7D%5Cright)%5Cleft(3-%5Csqrt%7B6%7D%5Cright)%7D%3D%5Cfrac%7B3%5Cleft(3-%5Csqrt%7B6%7D%5Cright)%7D%7B3%5E2-%5Csqrt%7B6%7D%5E2%7D%3D%5Cfrac%7B3%5Cleft(3-%5Csqrt%7B6%7D%5Cright)%7D%7B3%7D%3D3-%5Csqrt%7B6%7D" alt="\frac{3}{3+\sqrt{6}}=\frac{3\left(3-\sqrt{6}\right)}{\left(3+\sqrt{6}\right)\left(3-\sqrt{6}\right)}=\frac{3\left(3-\sqrt{6}\right)}{3^2-\sqrt{6}^2}=\frac{3\left(3-\sqrt{6}\right)}{3}=3-\sqrt{6}"/><br/> <br/> <br/> </span></div> 2026-09-29T09:39:14+03:00