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Question

Write a polynomial equation of least degree for each set of roots. Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis? -2, -0.5, 4

Solution

VerifiedAnswered 1 year ago

Answered 1 year ago

Step 1

1 of 2The corollary to the fundamental theorem of algebra states:

Every polynomial $P(x)$ with the degree $n$ ($n>0$) can be written as the product of a constant $k\neq 0$ and $n$ linear factors :

$P(x)=k(x-r_1)(x-r_2)\cdots(x-r_{n-1})(x-r_n)$

Thus a polynomial equation of degree $n$, has $n$ complex roots, namely $r_1,r_2,\cdots,r_n$.

Having $-2$,$-0.5$ and 4 as roots of a polynomial with the least degree means:

$P(x)=(x+2)(x-0.5)(x-4)$

The polynomial crosses the x-axis as many times as there are real roots, therefore 3 times.

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