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<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>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
To check the problem, you multiply
the divisor by the quotient and add the remainder to get the dividend.
remainder is 0, then
we say that the divisor <em>divides evenly</em> into the dividend.
<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>
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
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>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>
<p>When a polynomial function f is divided by x-k, the remainder r is f(k).
<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>Now, tie that into what we just said above.
If the remainder is zero, then you
factored the polynomial.
If the remainder when dividing by (x-k) is zero, then the
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
If the quotient is down to a quadratic or linear factor, then you can solve and
find the other