Imágenes de páginas
PDF
EPUB

EXERCISES

1. Prove by complete induction that the partial remainders up to the final remainder obtained in the process of synthetic division are the coefficients of the quotient of f(x) by x - α.

2. Perform by synthetic division the following divisions.

(a) x3 — 7 x2-6x + 72 ÷ x — 4.

[blocks in formation]

(g) x1 — 16 x3 + 86 x2 – 176 x + 105 ÷ x2 – 8 x + 7.

HINT. Since 2−8x+7=(x−7) (x − 1), divide by x-7 and the quotient by x-1.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

163. Plotting of equations. We can now form the table of values necessary to plot an equation of the type

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

In this figure two squares are taken to represent one unit of x. A single square represents a unit of y.

By an inspection of the figure it appears that the curve crosses the X axis at about x = .8, x = — - 1.2, and x = - 3.7. Thus the equation for y = 0 has approximately these values for roots (§ 110).

164. Extent of the table of values. Since the object of plotting a curve is to obtain information regarding the roots of its equation, stretches of the curve beyond all crossings of the X axis are of no interest for the present purpose. Hence it is desirable to know when a table of values has been formed extensive enough to afford a plot which includes all the real roots. If for all values of x greater than a certain number the curve lies wholly above the axis, there are no real roots greater than that value of x.

By inspection of the preceding example it appears that if for a given value of x the signs of the partial remainders are all positive, thus affording a positive value of y, any greater value of a will afford only positive partial remainders and hence only positive values of y.

Thus when all the partial remainders are positive no greater positive value of x need be substituted.

Similarly, when the partial remainders alternate in sign beginning with the coefficient of the highest power of x, no greater negative value of x need be substituted.

In plotting, if the table of values consists of values that are large or are so distributed that the plot would not be well proportioned if one space on the paper were taken for each unit, a scale should be so chosen that the plot will be of good proportion, that is, so that all the portions of the curve between the extreme roots shall appear on the paper, and the curvatures shall not be too abrupt to form a graceful curve. This was done, for example, in the figure, § 163.

EXERCISES

Plot and measure the values of the real roots of the equations when y = 0.

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

165. Roots of an equation. In the case of the linear and quadratic equations we have been able to find an explicit value of the roots in terms of the coefficients. Such processes are practically impossible in the case of most equations of higher degree. In fact the proof that any equation possesses a root lies beyond the scope of this book, and we make the

ASSUMPTION. Every equation possesses at least one root.

This is equivalent to the assumption that there is a number, rational, irrational, or complex, which satisfies any equation. 166. Number of roots. We determine the exact number of roots by the following

х

THEOREM. Every equation of degree n has n roots.

-1

...

Given the equation f(x)= a," + α1x2-1 + ··· + a1 = 0.

Let a1 (see assumption) be a root of this equation. Then (p. 166) xa is a factor of the left-hand member, and the quotient of f(x) by xa, is a polynomial of degree n-1. Suppose that а。x2 + α1x2−1+

-1

...

+ an

=

[ocr errors]

By our assumption the quotient an-1+b1n-2+ ··· + b2-1 = 0 has at least one root, say a2, to which corresponds the factor a2. Thus

[ocr errors]
[merged small][ocr errors][ocr errors][subsumed]

Proceeding in this way we find successive roots and corresponding linear factors until the polynomial is expressed as the product of n linear factors as follows:

[ocr errors]

ƒ (x) = α。 (x − α1) (x − α2) · · · (x − α„) = 0,

where the roots are a1, a2,

[ocr errors]
[ocr errors]

REMARK. This theorem gives no information regarding how many of the roots may be real or imaginary. This depends on the particular values of the coefficients.

COROLLARY. Any polynomial in x of degree n may be expressed as the product of n linear factors of the form xa, where a is a real or a complex number.

It should be noted that the roots are not necessarily distinct. Several of the roots and hence several of the factors may be identical.

If f(x) is divisible by (x — α1)2, that is, if a1 = a2, we say that a is a double root of the equation. Similarly, if f(x) is divisible by (xa1)", a, is called a multiple root of order r. When we say an equation has n roots we include each multiple root counted a number of times equal to its order.

THEOREM. An equation of degree n has no more than n distinct roots.

[merged small][merged small][merged small][merged small][subsumed][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors]

=

Since this numerical expression vanishes one of its factors must vanish (§ 5). But r‡ a1, thus r a10. Similarly, no one of the binomial factors vanishes. Thus (§ 5) a。 0, which contradicts the hypothesis that the equation is of degree n. This theorem may also be stated as follows:

ax

-1

COROLLARY I. If an equation аx2 + a1x2 -1 + ··· + a1 = 0 of degree n is satisfied by more than n values of x, all its coefficients vanish.

=

The proof of the theorem shows that if the equation has n + 1 roots, a 0. We should then have remaining an equation of degree n -1, also satisfied by n +1 values of x. Thus the coefficient of its highest power in x vanishes. Similarly, each of the coefficients vanishes.

COROLLARY II. If two polynomials in one variable are equal to each other for every value of the variable, the coefficients of like powers of the variable are equal and conversely.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

167. Graphical interpretation. The graphical interpretation of the theorems of the preceding section is that the graph of an equation of degree n cannot cross the X axis more than n times. Since each crossing of the X axis corresponds to a real root, there will be less than n crossings if the equation has imaginary roots. 168. Imaginary roots. We now show that imaginary roots occur in pairs. This we prove in the following

THEOREM. If a + ib is a root of an equation with real coefficients, aib is also a root of the equation.

If a + ib is a root of the equation a。x" + a1xn−1 +

then x

...

+ a2 = 0,

(a + b) is a factor (p. 166). We wish to prove that x ·(a — ib) is also a factor, or what amounts to the same thing, that their product

[x − (a + ib)] [x − (a — ib)] = [(x − a) — ib] [(x − a) + ib]

[blocks in formation]

is a factor of f(x). Divide ƒ(x) by (x − a)2 + b2 and we get

f(x)= Q(x)[(x − a)2 + b2] + rx + r',

(1)

where r and r' are real numbers. This remainder rx + r' can be of no higher degree in x than the first, since the divisor

[merged small][ocr errors]
« AnteriorContinuar »