College Algebra

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a: 4 + 3^ -4a; -10 = 0. 6. X s +20^-5^ + 7 = 0.
671. If two real numbers, a and b, not roots of the equation f{x) = x n +p 1 x"~ 1 + ... +p n _ 1 x +p n = 0, when substituted for x in f(x), give results of opposite sign, an odd number of roots off(x) = lie between a and b.
THEORY OF EQUATIONS. 483 Let a be algebraically greater than 6.
Let d, ..., g be the real roots of f(x) = lying between a and b, and h,...,k the remaining real roots.
Then by Art. 636, f(x) = (x-d)...{x-g).(x-h)...(x-k).F(x),
... (1) where F(x) is the product of the factors corresponding to the imaginary roots of f(x) = 0.
Substituting a and b for x in (1), we have (Art. 628), f(a) = (a-d)-(a-g)-(a-h)-(a-k)-F(a), and f(b)=(b-d)-(b-g)-(b-h)-(b-k)-F(b).
Since each of the quantities d, ..., g is less than a and greater than b, each of the factors a — d, ..., a — g is posi- tive, and each of the factors b — d, ..., b — g is negative.
Again, since none of the quantities h, ..., fc lie between a and &, the expression (a — 7i) • • • (a — fc) has the same sign as (b-h) — (b-k).


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