3.3 - Real Zeros of Polynomial Functions

3.3 - Real Zeros of Polynomial Functions - <?xml...

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<?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>3.3 - Real Zeros of Polynomial Functions</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>3.3 - Real Zeros of Polynomial Functions</h1> <h2>Long Division of Polynomials</h2> <p>You were taught long division of polynomials in Intermediate Algebra. Basically, the procedure is carried out like long division of real numbers. The procedure is explained in the textbook if you're not familiar with it. </p> <p>One key point about division, and this works for real numbers as well as for polynomial division, needs to be pointed out. When you divide the dividend by the divisor, you get a quotient and a remainder. To check the problem, you multiply the divisor by the quotient and add the remainder to get the dividend. If the remainder is 0, then we say that the divisor <em>divides evenly</em> into the dividend. </p> <p>Dividend / Divisor = Quotient + Remainder / Divisor</p> <p>Dividend = Divisor * Quotient + Remainder</p> <p>Like I said, the same thing can be done with polynomial functions.</p> <p> f(x) = d(x) * q(x) + r(x)</p> <p>Where f(x) is the polynomial function being divided into (dividend), d(x) is the polynomial function being divided by (divisor), q(x) is the polynomial function that is the quotient, and r(x) is the polynomial remainder function and will have degree less than the divisor. </p> <p>If the remainder, r(x), is zero, then f(x) = d(x)*q(x). We have just factored the function f(x) into two factors, d(x) and q(x).</p> <h2>Remainder Theorem</h2> <p>When a polynomial function f is divided by x-k, the remainder r is f(k). </p> <p>Okay, now in English. If you divide a polynomial by a linear factor, x-k, the remainder is the value you would get if you plugged x=k into the function and evaluated. </p> <p>Now, tie that into what we just said above. If the remainder is zero, then you have successfully factored the polynomial. If the remainder when dividing by (x-k) is zero, then the function evaluated at x=k is zero and you have found a zero or root of the polynomial. Plus, you now have a factored polynomial (the quotient) which is one less degree than the original polynomial. If the quotient is down to a quadratic or linear factor, then you can solve and find the other
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solutions.</p> <h2>Synthetic Division</h2> <p>To divide a polynomial synthetically by x-k, perform the following steps.</p> <h3>Setup</h3> <ol> <li>Write k down, leave some space after it.</li> <li>On the same line, right the coefficients of the polynomial function. Make sure you write the coefficients in order of decreasing power. Be sure to put a zero down if a power is missing. Place holders are very important</li>
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This note was uploaded on 10/18/2011 for the course MAT 1033 taught by Professor Brown during the Spring '10 term at Valencia.

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3.3 - Real Zeros of Polynomial Functions - &lt;?xml...

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