First Course in the Theory of Equations
Isolation of the Real Roots of a Real Equation
Excerpts
Isolation of the Real Roots of a Real Equation
But in the contrary case, narrower limits are necessary, such as $4$ and $4.3$, with the further fact that only one root is between these new limits. Then that root is said to be *isolated*.
Isolation of the Real Roots of a Real Equation
Thus, in $x^5 - 2x^3 - 4x^2 + 3 = 0$, the first two terms present a variation of sign, and likewise the last two terms. The number of variations of sign of the equation is two.
Isolation of the Real Roots of a Real Equation
The number of positive real roots of an equation with real coefficients is either equal to the number of its variations of sign or is less than that number by a positive even integer. A root of multiplicity $m$ is here counted as $m$ roots.
Isolation of the Real Roots of a Real Equation
For example, $x^6 - 3x^2 + x + 1 = 0$ has either two or no positive roots, the exact number not being found. But $3x^3 - x - 1 = 0$ has exactly one positive root, which is a simple root.
Isolation of the Real Roots of a Real Equation
Experience shows that most students make some error in finding $f_2, f_3, \dotsc$, so that checking is essential.
Isolation of the Real Roots of a Real Equation
Unfortunately it rarely tells us the exact number of real roots.
Isolation of the Real Roots of a Real Equation
A violation of this Corollary usually indicates an error in the computation of Sturm’s functions $(2)$.
Isolation of the Real Roots of a Real Equation
Descartes’ rule will be derived in §73 as a corollary to Budan’s theorem.
Equations
Isolation of the Real Roots of a Real Equation
f(x) \equiv a_0 x^n + a_1 x^{n-1} + \dotsb + a_l x^{n-l}Any real polynomial f(x) is written as a sum of descending powers of x with real coefficients, where the leading coefficient a_0 and the coefficient a_l are nonzero.
Isolation of the Real Roots of a Real Equation
F(x) \equiv (x-r)f(x) \equiv A_0 x^{n+1} + A_1 x^n + \dotsb + A_{l+1}x^{n-l}Multiplying f(x) by (x - r) gives a polynomial F(x) whose coefficients A_i are written in terms of the coefficients a_i of f.
Isolation of the Real Roots of a Real Equation
A_1 = a_1 - ra_0The second coefficient of F(x) equals a_1 minus r times a_0.
Isolation of the Real Roots of a Real Equation
A_{l+1} = -r a_lThe last coefficient of F(x) equals minus r times the last coefficient a_l of f(x).
Isolation of the Real Roots of a Real Equation
f(x) \equiv (x - r_1)\dotsm (x - r_k)\phi(x)A polynomial whose positive real roots are r_1 to r_k factors as the product of the linear factors (x - r_i) and a remaining polynomial phi(x).
Isolation of the Real Roots of a Real Equation
f = q_1 f_1 - f_2Dividing f by its derivative f_1 gives quotient q_1 and the negative of the remainder, which is the second function f_2 in Sturm's chain.
Isolation of the Real Roots of a Real Equation
f_{i-1}(x) = q_i f_i(x) - f_{i+1}(x)Each Sturm function is the quotient times the next Sturm function minus the one after it, the general step of the Sturm chain.
Isolation of the Real Roots of a Real Equation
f_{i-1}(\rho) = -f_{i+1}(\rho) \ne 0At a root rho of f_i, the neighbouring Sturm functions f_{i-1} and f_{i+1} take equal and opposite values, and that value is not zero.
Isolation of the Real Roots of a Real Equation
V_{r-p} - V_{r+p} = 1Passing a simple root r of f from left to right lowers the number of variations of sign of the Sturm sequence by exactly one.
Isolation of the Real Roots of a Real Equation
V_a\geqq V_{b}If a is less than b, the number of variations of sign at a is at least the number at b.
Isolation of the Real Roots of a Real Equation
c_{i+1} F_{i-1}(\rho) = -k_{i+1}(\rho) F_{i+1}(\rho)With the modified Sturm functions, at a root rho of F_i, F_{i-1} and F_{i+1} have opposite signs because their values are related by positive constants and a positive factor.
Isolation of the Real Roots of a Real Equation
f = z^4 + qz^2 + rz + sThe reduced quartic is defined with no cubic term, its coefficients named q, r, s.
Isolation of the Real Roots of a Real Equation
f_1 = 4z^3 + 2qz + rThe first derivative of the reduced quartic f with respect to z.
Isolation of the Real Roots of a Real Equation
f_2 = -2qz^2 - 3rz - 4sThe second Sturm function of the reduced quartic, the negative of the remainder of f divided by f_1.
Isolation of the Real Roots of a Real Equation
f_3 = Lz - 12rs - rq^2The third Sturm function of the reduced quartic is linear in z.
Isolation of the Real Roots of a Real Equation
L = 8qs - 2q^3 - 9r^2L is defined as 8qs minus 2q cubed minus 9r squared.
Isolation of the Real Roots of a Real Equation
\Delta = -4P^3 - 27Q^2The discriminant of the reduced quartic is expressed through the auxiliary quantities P and Q.
Isolation of the Real Roots of a Real Equation
P = -4s - \frac{q^2}{3}P is defined as minus 4s minus q squared over 3.
Isolation of the Real Roots of a Real Equation
Q = \tfrac{8}{3}qs - r^2 - \tfrac{2}{27}q^3Q is defined in terms of the coefficients q, r, s of the reduced quartic.
Isolation of the Real Roots of a Real Equation
4s = -P - \frac{q^2}{3}The constant term s can be eliminated: 4s equals minus P minus q squared over 3.
Isolation of the Real Roots of a Real Equation
r^2 = -Q - \tfrac{2}{3}qP - \tfrac{8}{27}q^3The square of the coefficient r can be written in terms of Q, P and q.
Isolation of the Real Roots of a Real Equation
f_3 = Lz + 3rPThe third Sturm function of the reduced quartic written with the alternative auxiliary quantities P and L.
Isolation of the Real Roots of a Real Equation
L = 9Q + 4qPL equals 9Q plus 4qP in terms of the auxiliary quantities Q and P.
Isolation of the Real Roots of a Real Equation
18r^2 qP^2 - 9r^2 LP + 4sL^2 = q^2 \DeltaThe negative of the remainder of L^2 f_2 on division by f_3 equals q^2 times the discriminant.
Isolation of the Real Roots of a Real Equation
f_4 = \DeltaThe last Sturm function of the reduced quartic is the (constant) discriminant, so the chain ends with the discriminant.
Isolation of the Real Roots of a Real Equation
V_0 - V_{\infty} = VThe number of variations of sign of f at x = 0 minus that at infinity equals V, the number of variations of sign of f(x), which gives Descartes' rule.
Isolation of the Real Roots of a Real Equation
f(x) \equiv a_0 x^n + a_1 x^{n-1} + \dotsb + a_{n-1}x + a_n = 0A general real polynomial equation of degree n with real coefficients and a_n nonzero, written with the constant term a_n.
Problems
Exercise Page74
Exercise Page74, problem 1, p. 74
An equation all of whose coefficients are of like sign has no positive root. Why is this self-evident?
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Exercise Page74, problem 10, p. 74
Prove that we obtain an upper limit to the number of real roots of $f(x)=0$ between $a$ and $b$, if we set x = a+by1+y (∴y=x-ab-x) multiply by $(1+y)^n$, and apply Descartes’ rule to the resulting equation in $y$.
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Exercise Page74, problem 11, p. 74
Show by the method of Ex. 10 that there is a single root between $2$ and $4$ of $x^3 + x^2 - 17x + 15 = 0$. Here we have $27y^3 + 3y^2 - 23y - 7 = 0$.
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Exercise Page74, problem 12, p. 74
In the astronomical problem of three bodies occurs the equation r^5 + (3 - )r^4 + (3 - 2)r^3 - r^2 - 2r - = 0, where $0 < \mu < 1$. Why is there a single positive real root?
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Exercise Page74, problem 13, p. 74
Prove that $x^5 + x^3 - x^2 + 2x - 3 = 0$ has four imaginary roots by applying Descartes’ rule to the equation in $y$ whose roots are the squares of the roots of the former. Transpose the odd powers, square each new member, and replace $x^2$ by $y$.
Printed answer:- $y^5 + 2y^4 + 5y^3 + 3y^2 - 2y - 9 = 0$.
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Exercise Page74, problem 14, p. 74
As in Ex. 13 prove that $x^3 + x^2 + 8x + 6 = 0$ has imaginary roots.
Printed answer:- $y^3 + 15y^2 + 52y - 36 = 0$.
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Exercise Page74, problem 15, p. 74
If a real equation $f(x)=0$ of degree $n$ has $n$ real roots, the number of positive roots is exactly equal to the number $V$ of variations of sign. Hint: consider also $f(-x)$.
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Exercise Page74, problem 16, p. 74
Show that $x^3 - x^2 + 2x + 1 = 0$ has no positive root. Hint: multiply by $x + 1$.
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Exercise Page74, problem 2, p. 74
There is no negative root of an equation, like $x^5 - 2x^4 - 3x^2 + 7x - 5 = 0$, in which the coefficients of the odd powers of $x$ are of like sign, and the coefficients of the even powers (including the constant term) are of the opposite sign. Verify by taking $x= -p$, where $p$ is positive.
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Exercise Page74, problem 3, p. 74
$x^3 + a^2 x + b^2 = 0$ has two imaginary roots if $b\ne 0$.
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Exercise Page74, problem 4, p. 74
For $n$ even, $x^n - 1 = 0$ has only two real roots.
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Exercise Page74, problem 5, p. 74
For $n$ odd, $x^n - 1 = 0$ has only one real root.
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Exercise Page74, problem 6, p. 74
For $n$ even, $x^n + 1 = 0$ has no real root; for $n$ odd, only one.
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Exercise Page74, problem 7, p. 74
$x^4 + 12x^2 + 5x - 9 = 0$ has just two imaginary roots.
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Exercise Page74, problem 8, p. 74
$x^4 + a^2 x^2 + b^2 x - c^2 = 0$ ($c\ne 0$) has just two imaginary roots.
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Exercise Page74, problem 9, p. 74
Descartes’ rule enables us to find the exact number of positive roots only when all the coefficients are of like sign or when f(x) = x^n + p_1 x^n-1 + + p_n-s x^s - p_n-s+1 x^s-1 - - p_n = 0, each $p_i$ being $\geqq 0$. Without using that rule, show that the latter equation has one and only one positive root $r$. Hints: There is a positive root $r$ by §63 ($a=0$, $b=\infty$). Denote by $P(x)$ the quotient of the sum of the positive terms by $x^s$, and by $-N(x)$ that of the negative terms. Then $N(x)$ is a sum of powers of $1/x$ with positive coefficients. align* If x>r, P(x)>P(r), N(x)<N(r), f(x)>0; If x<r, P(x)<P(r), N(x)>N(r), f(x)<0. align*
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Exercise Page78
Exercise Page78, problem 1, p. 78
$x^3 +2x +20 = 0$.
Printed answer:- One, between $-2$ and $-3$.
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Exercise Page78, problem 2, p. 78
$x^3 +x-3 = 0$.
Printed answer:- One, between $1$ and $2$.
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Exercise Page79
Exercise Page79, problem 1, p. 79
$x^3 + 3x^2 - 2x - 5 = 0$.
Printed answer:- $(-4, -3)$, $(-2, -1)$, $(1, 2)$.
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Exercise Page79, problem 2, p. 79
$x^4 + 12x^2 + 5x - 9 = 0$.
Printed answer:- $(-2, -1)$, $(0, 1)$.
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Exercise Page79, problem 3, p. 79
$x^3 - 7x - 7 = 0$.
Printed answer:- $(-2, -1.5)$, $(-1.5, -1)$, $(3, 4)$.
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Exercise Page79, problem 4, p. 79
$3x^4 - 6x^2 + 8x - 3 = 0$.
Printed answer:- $(-2, -1)$, $(0, 1)$.
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Exercise Page79, problem 5, p. 79
$x^6 + 6x^5 - 30x^2 - 12x - 9 = 0$ [stop with $f_2$].
Printed answer:- $(-7, -6)$, $(1, 2)$.
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Exercise Page79, problem 6, p. 79
$x^4 - 8x^3 + 25x^2 - 36x + 8 = 0$.
Printed answer:- $(0, 1)$, $(3, 4)$.
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Exercise Page79, problem 7, p. 79
For $f = x^3 + px + q$ ($p\ne 0$), show that $f_1 = 3x^2 + p$, $f_2 = -2px - 3q$, 4p^2 f_1 = (-6px + 9q)f_2 - f_3, f_3 = -4p^3 - 27q^2, so that $f_3$ is the discriminant $\Delta$ (§44). Let $[p]$ denote the sign of $p$. Then the signs of $f$, $f_1$, $f_2$, $f_3$ are align* &-++ [p] [] for $x = -\infty$, &++- [p] [] for $x = +\infty$. align* For $\Delta$ negative there is a single real root. For $\Delta$ positive and therefore $p$ negative, there are three distinct real roots. For $\Delta = 0$, $f_2$ is a divisor of $f_1$ and $f$, so that $x = -3q/(2p)$ is a double root.
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Exercise Page79, problem 8, p. 79
Prove that if one of Sturm’s functions has $p$ imaginary roots, the initial equation has at least $p$ imaginary roots.
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Exercise Page79, problem 9, p. 79
State Sturm’s theorem so as to include the possibility of $a$, or $b$, or both $a$ and $b$ being roots of $f(x)=0$.
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Exercise Page81
The data holds no problems for this exercise yet.
Exercise Page83
Exercise Page83, problem 1, p. 83
For $f = x^4 - 8x^2 + 16$, prove that $F_1 = x^3 - 4x$, $F_2 = x^2 - 4$, $F_1 = xF_2$. Hence $n = 2$. Verify that $V_{-\infty} = 2$, $V_{\infty} = 0$, and that there are just two real roots, each a double root.
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Exercise Page83, problem 2, p. 83
$x^4 - 5x^3 + 9x^2 - 7x + 2 = 0$.
Printed answer:- $1$, $1$, $1$, $2$.
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Exercise Page83, problem 3, p. 83
$x^4 + 2x^3 - 3x^2 - 4x + 4 = 0$.
Printed answer:- $1$, $1$, $-2$, $-2$.
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solve: passes[1, 1, -2, -2]
Exercise Page83, problem 4, p. 83
$x^4 - x^2 - 2x + 2 = 0$.
Printed answer:- $1$, $1$, two imaginary.
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Exercise Page85
Exercise Page85, problem 1, p. 85
$x^3 -x^2 -2x+1=0$.
Printed answer:- $(-2, -1)$, $(0, 1)$, $(1, 2)$.
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Exercise Page85, problem 2, p. 85
$x^3 +3x^2 -2x-5=0$.
Printed answer:- $(-4, -3)$, $(-2, -1)$, $(1, 2)$.
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Exercise Page85, problem 3, p. 85
Prove that if $f{(a)}\ne 0$, $V_a$ equals the number of real roots $>a$ or exceeds that number by an even integer.
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Exercise Page85, problem 4, p. 85
Prove that there is no root greater than a number making each of the functions $(12)$ positive, if the leading coefficient of $f(x)$ is positive. (Newton.)
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Exercise Page85, problem 5, p. 85
Hence verify that $x^4 -4x^3 - 3x + 23 = 0$ has no root $>4$.
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Exercise Page85, problem 6, p. 85
Show that $x^4 - 4x^3 + x^2 + 6x + 2 = 0$ has no root $>3$.
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