https://peda.net/id/2038891a2cf 2022-09-05T12:42:41+03:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/203f4c6c2cf 2020-11-30T17:39:57+02:00 <p class="p1">When <b>polynomials</b> are multiplied, all terms inside the brackets must be multiplied separately. Similarly, when dividing a polynomial, each term must be divided separately.</p> <h3><b>Example 1</b></h3> <p class="p1">Calculate the division [[$ \displaystyle\frac {16x^2 - 4x} {4x} $]].​</p> <p class="p1"><b>Method I </b></p> <p class="p1">Divide each term of the polynomial [[$ 16x^2 - 4x $]] separately by the monomial [[$ 4x $]].<br/> <br/> [[$ \displaystyle\frac {16x^2 - 4x} {4x} = \displaystyle\frac {16x^2} {4x} - \displaystyle\frac {4x} {4x} = 4x - 1 $]]​</p> <p class="p1"><b>Method II </b></p> <p class="p1">The division calculation can also be accomplished by first converting the division into product form and then reducing it by the common terms of the numerator and denominator.<br/> <br/> <span><a href="https://peda.net/qis/2022-2023/mathematics/matematiikka-92/oikjs/1pjmjp/1pjmjp/1#top" title="10_dividing-polynomials-example1.png"><img src="https://peda.net/qis/2022-2023/mathematics/matematiikka-92/oikjs/1pjmjp/1pjmjp/1:file/photo/ba2b3859d8cf5f0162de3d7fc0f30236c5d72ee4/10_dividing-polynomials-example1.png" alt="" title="The numerator is divided into factors. Common factors can be reduced, as the numerator is presented in product form." class="inline" loading="lazy"/></a> </span></p> <p class="p1"><b>Note! </b>If a polynomial that contains a multiplication is to be divided, only the multiplier and the multiplicand can be divided. If a division calculation were made for both factors, the division calculation should be performed twice.</p> <div class="eoppi-summary"> <h3><b>Dividing a polynomial by a monomial</b></h3> <ul> <li>If the dividend is kept in the <b>sum form</b>, each term must be divided separately.</li> <li>If the dividend is changed to the <b>product form</b>, only either the multiplier or the multiplicand is divided, but not both.</li> </ul> </div> <p class="p1">If the polynomial is divided by a polynomial with at least two terms, the first method shown in Example 1 cannot be used. Simplifying such a fraction always requires dividing the numerator and denominator into product form parts that can be reduced.</p> <p class="p2"><span class="s1">The most common mistake when simplifying rational expressions containing polynomials is to remove individual terms from a polynomial that is presented in the sum form. In this case, no division calculation has been performed for each term to be divided. The sum should never be reduced!</span></p> <h3><b>Example 2</b></h3> <p class="p1">When a polynomial is divided by a polynomial, the expressions must first be displayed in product form.<br/> <br/> <a href="https://peda.net/qis/2022-2023/mathematics/matematiikka-92/oikjs/1pjmjp/1pjmjp/12#top" title="10_dividing-polynomials-example2.png"><img src="https://peda.net/qis/2022-2023/mathematics/matematiikka-92/oikjs/1pjmjp/1pjmjp/12:file/photo/9efa413c94b605f36911c0c7767dee32c9c920b5/10_dividing-polynomials-example2.png" alt="" title="The numerator and denominator are divided into factors. Can be reduced, as both the numerator and denominator are presented in product form. The numerator and denominator are given a common factor by multiplying (-x+2) with the number -1." class="inline" loading="lazy"/></a> <br/> <b>Note! </b>If -2 had been chosen as the common factor of the denominator, a common factor for the denominator and numerator could have been obtained immediately.</p> 2022-09-05T12:42:41+03:00