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60 The Nuts and Bolts of Proof, Third Edition
Proof:
1. The point-slope equation of a line is:
y - yo = m(x - XQ)
where m is the slope and (XQ, yo) are the coordinates of any point on the
Hne.
If (^i,yi) and (xo?}^o) are the coordinates of any two points on the line,
and x\ ^ xo, then:
m = .
Xi -Xo
Using the points with coordinates (0,2) and (2,6), we obtain
m=:4/2 = 2.
Therefore, the Hne that passes through the given points has the
equation:
or
y-2 = 2(x-0)
y = 2x-\-2.
m
2. There is a postulate from geometry that states that given any
two distinct points in the plane there is a unique straight Hne joining
them. Therefore, there is a Hne joining the points with coordinates
(0,2) and (2,6). •
EXAMPLE 4. The polynomial P{x) = x^ -^ x^ -{-x^ -\- x— 1 has a real zero
in the interval [0,1].
Discussion: If we want to find an expHcit value of x such that P(x) = 0, we
need to solve a fourth-degree equation. This can be done, but the formulas
used to solve a fourth-degree equation are quite cumbersome, even if they
are not difficult. It is possible to use a calculator, or any numerical method
(such as Newton's method) as well, but the statement does not ask us to find
the real zeroes of the polynomial P(x). We are only asked to prove that one
of the zeroes is in the interval [0,1]. The proof that foHows requires some
knowledge of calculus.
Proof. Polynomials are continuous functions. Because P(0) = — 1 is a
negative number and P(l) = 3 is a positive number, by the intermediate
value theorem there will be at least one value of x in the interval [0,1] for
which P{x) = 0. Thus, the given statement is true. •