Elementary Theorems on the Roots of an Equation

Excerpts

Elementary Theorems on the Roots of an Equation
is called a *quadratic equation* or equation of the second degree.
Elementary Theorems on the Roots of an Equation
If a polynomial $f(x)$ be divided by $x - c$ until a remainder independent of $x$ is obtained, this remainder is equal to $f(c)$, which is the value of $f(x)$ when $x = c$.
Elementary Theorems on the Roots of an Equation
If $f(c)$ is zero, the polynomial $f(x)$ has the factor $x - c$. In other words, if $c$ is a root of $f(x) = 0$, $x - c$ is a factor of $f(x)$.
Elementary Theorems on the Roots of an Equation
First we bring down the first coefficient $1$. Then we multiply it by the given value $2$ and enter the product $2$ directly under the second coefficient $3$, add and write the sum $5$ below.
Elementary Theorems on the Roots of an Equation
We thus obtain the useful result that $ax^2 + bx + c$ *is a perfect square (of a linear function of $x$) if and only if $b^2 = 4ac$ (*i.e.*, if its discriminant is zero)*.
Elementary Theorems on the Roots of an Equation
An equation of degree $n$ cannot have more than $n$ roots, a root of multiplicity $m$ being counted as $m$ roots.
Elementary Theorems on the Roots of an Equation
For example, in $x^5 + 4x^4 - 7x^2 - 40x + 1 = 0$, $G = 40$ and $k = 3$ since we must supply the coefficient zero to the missing power $x^3$.

Equations

Elementary Theorems on the Roots of an Equation
ax^2 + bx + c = 0 \quad (a \ne 0)
The general quadratic equation, with a nonzero leading coefficient a, is called a quadratic equation or equation of the second degree.
Elementary Theorems on the Roots of an Equation
(2ax + b)^2 = \Delta
Completing the square turns the quadratic into a square equal to the discriminant Delta.
Elementary Theorems on the Roots of an Equation
x_{1} = \frac{-b + \sqrt{\Delta}}{2a}
The first root of the quadratic equation is given by the quadratic formula with the plus sign on the square root of the discriminant.
Elementary Theorems on the Roots of an Equation
x_{2} = \frac{-b - \sqrt{\Delta}}{2a}
The second root of the quadratic equation is given by the quadratic formula with the minus sign on the square root of the discriminant.
Elementary Theorems on the Roots of an Equation
x_{1} + x_{2} = \frac{-b}{a}
The sum of the two roots of a quadratic equation is minus b over a.
Elementary Theorems on the Roots of an Equation
x_{1} x_{2} = \frac{ c}{a}
The product of the two roots of a quadratic equation is c over a.
Elementary Theorems on the Roots of an Equation
a(x - x_1)(x - x_2) \equiv ax^2 - a(x_1 + x_2)x + ax_1 x_2 \equiv ax^2 + bx + c
For all x, the quadratic factors as a(x - x_1)(x - x_2), and this is identically equal to ax^2 + bx + c (the factored form).
Elementary Theorems on the Roots of an Equation
0 = ax_1^2 + bx_1 + c
Substituting the first root x_1 into the quadratic gives zero, so x_1 is a root of the equation.
Elementary Theorems on the Roots of an Equation
0 = ax_2^2 + bx_2 + c
Substituting the second root x_2 into the quadratic gives zero, so x_2 is a root of the equation.
Elementary Theorems on the Roots of an Equation
\Delta = b^2 - 4ac
The discriminant Delta of the quadratic function or equation is b^2 - 4ac.
Elementary Theorems on the Roots of an Equation
b^2 = 4ac
The quadratic ax^2 + bx + c is a perfect square of a linear function of x if and only if its discriminant is zero, b^2 = 4ac.
Elementary Theorems on the Roots of an Equation
f(x) \equiv c_0 x^n + c_1 x^{n-1} + \dotsb + c_{n-1} x + c_n
A polynomial of degree n in x is a sum of constant multiples of successive powers of x, also called an integral rational function of degree n.
Elementary Theorems on the Roots of an Equation
f(x) \equiv (x-c)q(x) + r
Dividing f(x) by x - c leaves a constant remainder r and quotient q(x), identically in x.
Elementary Theorems on the Roots of an Equation
f(c) = r
The remainder on dividing a polynomial f(x) by x - c equals f(c), the value of f at x = c.
Elementary Theorems on the Roots of an Equation
b_1 = a_1 + cb_0
In synthetic division by x - c, each new coefficient of the quotient is the next coefficient of f plus c times the previous quotient coefficient.
Elementary Theorems on the Roots of an Equation
r = a_n + cb_{n-1}
In synthetic division by x - c, the remainder is the constant coefficient of f plus c times the last quotient coefficient.
Elementary Theorems on the Roots of an Equation
f(x) \equiv (x - \alpha_1)Q(x)
If alpha_1 is a root of f(x) = 0, then x - alpha_1 is a factor, so f(x) equals (x - alpha_1) times a polynomial Q(x) of degree one less.
Elementary Theorems on the Roots of an Equation
f(x) \equiv c_0(x - \alpha_1)(x - \alpha_2) \dotsm (x - \alpha_n)
A polynomial of degree n with leading coefficient c_0 is a product of n linear factors built from its roots (the factored form).
Elementary Theorems on the Roots of an Equation
f(x) \equiv c_0(x-\alpha_1)^{m_1} (x-\alpha_2)^{m_2} \dotsm (x-\alpha_k)^{m_k}, \quad m_1 + m_2 + \dotsb + m_k = n
With distinct roots alpha_1 to alpha_k of multiplicities m_1 to m_k, the polynomial factors with each linear factor raised to its multiplicity, and the multiplicities sum to the degree n.
Elementary Theorems on the Roots of an Equation
a_0 = b_0
Two polynomials of degree n equal in value at more than n distinct points are term by term identical, so their leading coefficients agree (and likewise all coefficients).
Elementary Theorems on the Roots of an Equation
f(x) \equiv c_0 x^n + c_1 x^{n-1} + \dotsb + c_n = 0\qquad (c_0 \ne 0)
An equation f(x) = 0 of degree n with nonzero leading coefficient c_0 is the general polynomial equation of degree n.
Elementary Theorems on the Roots of an Equation
(x - \alpha_1)(x - \alpha_2) &\equiv x^2 - (\alpha_1 + \alpha_2)x + \alpha_1\alpha_2
The product of two linear factors expands to a quadratic whose x coefficient is minus the sum of the roots and whose constant is their product.
Elementary Theorems on the Roots of an Equation
(x - \alpha_1)(x - \alpha_2) \dotsm (x - \alpha_n) \equiv x^n - (\alpha_1 + \dotsb + \alpha_n)x^{n-1} \\ + (\alpha_1\alpha_2 + \alpha_1\alpha_3 + \alpha_2\alpha_3 + \dotsb + \alpha_{n-1}\alpha_n)x^{n-2} \\ - (\alpha_1\alpha_2\alpha_3 + \alpha_1\alpha_2\alpha_4 + \dotsb + \alpha_{n-2}\alpha_{n-1}\alpha_n)x^{n-3} \\ + \dotsb + (-1)^n \alpha_1\alpha_2 \dotsm \alpha_n
The product of n linear factors expands with coefficients given by the elementary symmetric functions of the roots, with alternating signs; proved by mathematical induction.
Elementary Theorems on the Roots of an Equation
\alpha_1 + \alpha_2 + \dotsb + \alpha_n &= -c_1 / c_0
The sum of the roots of the equation equals minus c_1 over c_0, the negative of the coefficient of the second term over the leading coefficient.
Elementary Theorems on the Roots of an Equation
\alpha_1\alpha_2 \dotsm \alpha_{n-1}\alpha_n &= (-1)^n c_n / c_0
The product of all the roots equals (-1)^n times c_n over c_0, the constant term over the leading coefficient with sign.
Elementary Theorems on the Roots of an Equation
(x-a)^2 + b^2 \equiv (x - a-bi)(x - a+bi)
The real quadratic (x - a)^2 + b^2 factors into the two conjugate complex linear factors x - (a + bi) and x - (a - bi).
Elementary Theorems on the Roots of an Equation
f(x) \equiv Q(x)\bigl\{(x-a)^2 + b^2\bigr\} + rx + s
Dividing a real polynomial by (x - a)^2 + b^2 leaves a quotient Q(x) and a linear remainder rx + s, identically in x.
Elementary Theorems on the Roots of an Equation
x \geqq 1 + \sqrt[k]{G / a_0}
Any real x at least 1 plus the k-th root of G over a_0 is not a root, so that bound is an upper limit to the real roots.
Elementary Theorems on the Roots of an Equation
x^{n-k} + \dotsb + x + 1 \equiv \frac{x^{n-k+1} - 1}{x - 1}
The geometric sum of successive powers of x from 0 to n-k equals (x^{n-k+1} - 1) over (x - 1), for x not equal to 1.
Elementary Theorems on the Roots of an Equation
x^4 \equiv (x-1) (x^3 + x^2 + x + 1) + 1
The fourth power of x is written as (x - 1) times a cubic in x plus 1, an identity used in the proof of Theorem II.
Elementary Theorems on the Roots of an Equation
x^2 \equiv (x-1) (x+1) + 1
The square of x is written as (x - 1)(x + 1) plus 1, an identity used in the proof of Theorem II.
Elementary Theorems on the Roots of an Equation
x \geqq 1 + \frac{-a_{k_i}}{\sum a_m}
For each negative coefficient a_{k_i}, any x at least 1 plus minus a_{k_i} over the sum of the preceding positive coefficients is not a root; the greatest such bound is an upper limit to the roots.
Elementary Theorems on the Roots of an Equation
x \geqq 1 + \frac{-a_0}{\sum a_m}
When the constant term a_0 is negative, any x at least 1 plus minus a_0 over the sum of the positive coefficients is not a root.
Elementary Theorems on the Roots of an Equation
a_0 x^n + \dotsb + a_{n-1}x + a_n = 0
A polynomial equation of degree n in the unknown x, whose coefficients a_0, ..., a_n are integers, is set equal to zero; this is the equation the integral-root theorem is applied to.
Elementary Theorems on the Roots of an Equation
f(x) \equiv x^4 - 9x^3 + 24x^2 - 23x + 15 = 0
The polynomial f(x) is defined as the quartic x^4 - 9x^3 + 24x^2 - 23x + 15, and the equation f(x) = 0 is the one whose roots are sought.
Elementary Theorems on the Roots of an Equation
15y^4 - 23y^3 + 24y^2 - 9y + 1 = 0
Replacing x by 1/y in f(x) = 0 and multiplying through by y^4 gives this equation in y, the transformed equation whose root y = 1/3 corresponds to the root x = 3.
Elementary Theorems on the Roots of an Equation
c_0 x^n + c_1 x^{n-1} + \dotsb + c_{n-1} x + c_n = 0
The general equation of degree n with integral coefficients c_0, ..., c_n, to which the rational-root theorem applies.
Elementary Theorems on the Roots of an Equation
f(x) \equiv (x-d)Q(x)
If d is a root of f(x) = 0, then f(x) is identically the product of (x - d) and a polynomial Q(x) with integral coefficients.
Elementary Theorems on the Roots of an Equation
f(m) = (m-d) q
For an integer m, the value f(m) is divisible by m - d whenever d is a root, because q = Q(m) is an integer; this gives the test used to exclude candidate roots.

Problems

Exercise Page25

Exercise Page25, problem 1, p. 25
$x^3 + 8x^2 + 13x + 6 = 0$.
Printed answer:
- $-1$, $-1$, $-6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-1, -1, -6]
Exercise Page25, problem 2, p. 25
$x^3 - 5x^2 - 2x + 24 = 0$.
Printed answer:
- $-2$, $3$, $4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-2, 3, 4]
Exercise Page25, problem 3, p. 25
$x^3 - 10x^2 + 27x - 18 = 0$.
Printed answer:
- $1$, $3$, $6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, 3, 6]
Exercise Page25, problem 4, p. 25
$x^4 + 4x^3 + 8x + 32 = 0$.
Printed answer:
- $-2$, $-4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-2, -4]
Exercise Page25, problem 5, p. 25
The equation in Ex. 4 of §23.
Printed answer:
- None.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page26

The data holds no problems for this exercise yet.

Exercise Page27

Exercise Page27, problem 1, p. 27
$x^4 - 2x^3 - 21x^2 + 22x + 40 = 0$.
Printed answer:
- $2$, $-1$, $-4$, $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, -1, -4, 5]
Exercise Page27, problem 2, p. 27
$y^3 - 9y^2 - 24y + 216 = 0$.
Printed answer:
- $9$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: FLAG-PARSE 9
Exercise Page27, problem 3, p. 27
$x^4 - 23x^3 + 187x^2 - 653x + 936 = 0$.
Printed answer:
- $8$, $9$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [8, 9]
Exercise Page27, problem 4, p. 27
$x^5 + 47x^4 + 423x^3 + 140x^2 + 1213x - 420 = 0$.
Printed answer:
- $-12$, $-35$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-12, -35]
Exercise Page27, problem 5, p. 27
$x^5 - 34x^3 + 29x^2 + 212x - 300 = 0$.
Printed answer:
- $2$, $2$, $-3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, 2, -3]

Exercise Page28

Exercise Page28, problem 1, p. 28
$y^4 -\frac{40}{3}y^3 + \frac{130}{3}y^2 - 40y + 9 = 0$.
Printed answer:
- $1$, $3$, $9$, $\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, 3, 9, Rational(1,3)]
Exercise Page28, problem 10, p. 28
$y^2 - 2y - \frac{1}{3} = 0$.
Printed answer:
- $x^2 - 12x - 12 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page28, problem 11, p. 28
$y^3 - \frac{1}{2}y^2 - \frac{1}{3}y + \frac{1}{4} = 0$.
Printed answer:
- $x^3 - 3x^2 - 12x + 54 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page28, problem 2, p. 28
$6y^3 - 11y^2 + 6y - 1 = 0$.
Printed answer:
- $1$, $\tfrac{1}{2}$, $\tfrac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, Rational(1,2), Rational(1,3)]
Exercise Page28, problem 3, p. 28
$108y^3 - 270y^2 - 42y + 1 = 0$. [Use $k = 6$.]
Printed answer:
- $-\tfrac{1}{6}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes Rational(-1,6)
On the STU-32 (STU, rpn):
```
1 +/− ENTER 6 ÷
```
Calculator: -1666666666666666666666666666666667E-34; the book prints -1/6. Run on the calculator core at firmware 628c96c.
Exercise Page28, problem 4, p. 28
$32y^3 - 6y - 1 = 0$. [Use the least $k$.]
Printed answer:
- $\tfrac{1}{2}$, $-\tfrac{1}{4}$, $-\tfrac{1}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(1,2), Rational(-1,4), Rational(-1,4)]
Exercise Page28, problem 5, p. 28
$96y^3 - 16y^2 - 6y + 1 = 0$.
Printed answer:
- $\tfrac{1}{4}$, $-\tfrac{1}{4}$, $\tfrac{1}{6}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(1,4), Rational(-1,4), Rational(1,6)]
Exercise Page28, problem 6, p. 28
$24y^3 - 2y^2 - 5y + 1 = 0$.
Printed answer:
- $-\tfrac{1}{2}$, $\tfrac{1}{3}$, $\tfrac{1}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(-1,2), Rational(1,3), Rational(1,4)]
Exercise Page28, problem 7, p. 28
$y^3 - \frac{1}{2}y^2 - 2y + 1 = 0$.
Printed answer:
- $\tfrac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes Rational(1,2)
On the STU-32 (STU, rpn):
```
1 ENTER 2 ÷
```
```
1 ENTER 2 ÷ −
```
```
1 ENTER 2 ÷ ×
```
```
2 −
```
```
1 ENTER 2 ÷ ×
```
```
1 +
```
```
1 ENTER 2 ÷
```
Calculator: +5E-1; the book prints 1/2. Run on the calculator core at firmware 628c96c.
Exercise Page28, problem 8, p. 28
$y^3 - \frac{2}{3}y^2 + 3y - 2 = 0$.
Printed answer:
- $\tfrac{2}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes Rational(2,3)
On the STU-32 (STU, rpn):
```
2 ENTER 3 ÷ ENTER 2 ENTER 3 ÷ −
```
```
× 3 +
```
```
2 ENTER 3 ÷ × 2 −
```
```
2 ENTER 3 ÷
```
Calculator: +6666666666666666666666666666666667E-34; the book prints 2/3. Run on the calculator core at firmware 628c96c.
Exercise Page28, problem 9, p. 28
Solve Exs. 2--6 by replacing $y$ by $1/x$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page13

Exercise Page13, problem 1, p. 13
$x^4 - 3x^2 - x - 6$ is divided by $x + 3$.
Printed answer:
- $51$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 51
On the STU-32 (STU, rpn):
```
3 +/− GOLD x²
```
```
ENTER GOLD x²
```
```
x↔y 3 ×
```
```
− 3 +
```
```
6 −
```
Calculator: +51E+0; the book prints 51. Run on the calculator core at firmware 628c96c.
Exercise Page13, problem 10, p. 13
If $a$, $ar$, $ar^2, \dotsc, ar^{n-1}$ are $n$ numbers in *geometrical progression* (the ratio of any term to the preceding being a constant $r \ne 1$), prove by Exercise 7 that their sum is equal to % a(r^n - 1)r - 1.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 11, p. 13
At the end of each of $n$ years a man deposits in a savings bank $a$ dollars. With annual compound interest at 4%, show that his account at the end of $n$ years will be % a.04 (1.04)^n - 1 dollars. Hint: The final deposit draws no interest; the prior deposit will amount to $a(1.04)$ dollars; the deposit preceding that will amount to $a(1.04)^2$ dollars, etc. Hence apply Exercise 10 for $r = 1.04$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 2, p. 13
$x^3 - 3x^2 + 6x - 5$ is divided by $x - 3$.
Printed answer:
- $13$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 13
On the STU-32 (STU, rpn):
```
3 ENTER 3 yˣ
```
```
3 ENTER GOLD x² ×
```
```
−
```
```
6 ENTER 3 × +
```
```
5 −
```
Calculator: +1300000000000000000000000000000000E-32; the book prints 13. Run on the calculator core at firmware 628c96c.
Exercise Page13, problem 3, p. 13
$18x^{10} + 19x^5 + 1$ is divisible by $x + 1$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 4, p. 13
$2x^4 - x^3 - 6x^2 + 4x - 8$ is divisible by $x - 2$ and $x + 2$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 5, p. 13
$x^4 - 3x^3 + 3x^2 - 3x + 2$ is divisible by $x - 1$ and $x - 2$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 6, p. 13
$r^3 - 1$, $r^4 - 1$, $r^5 - 1$ are divisible by $r - 1$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 7, p. 13
By performing the indicated multiplication, verify that r^n - 1 (r - 1)(r^n-1 + r^n-2 + + r + 1).
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 8, p. 13
In the last identity replace $r$ by $x/y$, multiply by $y^n$, and derive x^n - y^n (x-y)(x^n-1 + x^n-2y + + xy^n-2 + y^n-1).
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page13, problem 9, p. 13
In the identity of Exercise 8 replace $y$ by $-y$, and derive align* x^n + y^n &(x+y)(x^n-1 - x^n-2 y + - xy^n-2 + y^n-1), $n$ odd; x^n - y^n &(x+y)(x^n-1 - x^n-2 y + + xy^n-2 - y^n-1), $n$ even. align*
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page15

Exercise Page15, problem 1, p. 15
Divide $x^3 + 3x^2 - 2x - 5$ by $x-2$.
Printed answer:
- Rem. $11$, quot. $x^2 + 5x + 8$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page15, problem 2, p. 15
Divide $2x^5 - x^3 + 2x - 1$ by $x+2$.
Printed answer:
- $-61$, $2x^4 - 4x^3 + 7x^2 - 14x + 30$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page15, problem 3, p. 15
Divide $x^3 + 6x^2 + 10x - 1$ by $x - 0.09$.
Printed answer:
- $-0.050671$, $x^2 + 6.09x + 10.5481$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page15, problem 4, p. 15
Find the quotient of $x^3 - 5x^2 - 2x + 24$ by $x-4$, and then divide the quotient by $x-3$. What are the roots of $x^3 - 5x^2 - 2x + 24 = 0$?
Printed answer:
- $x^2 - x - 6$, $x+2$; $4$, $3$, $-2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page15, problem 5, p. 15
Given that $x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$ has the roots $-1$ and $2$, find the quadratic equation whose roots are the remaining two roots of the given equation, and find these roots.
Printed answer:
- $x^2 - x - 6 = 0$, $3$, $-2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page15, problem 6, p. 15
If $x^4 - 2x^3 - 12x^2 + 10x + 3 = 0$ has the roots $1$ and $-3$, find the remaining two roots.
Printed answer:
- $2±\sqrt{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2 - sqrt(5), 2 + sqrt(5)]
Exercise Page15, problem 7, p. 15
Find the quotient of $2x^4 - x^3 - 6x^2 + 4x - 8$ by $x^2 - 4$.
Printed answer:
- $2x^2 - x + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2 - x + 2
Exercise Page15, problem 8, p. 15
Find the quotient of $x^4 - 3x^3 + 3x^2 - 3x + 2$ by $x^2 - 3x + 2$.
Printed answer:
- $x^2 + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 1
Exercise Page15, problem 9, p. 15
Solve Exercises 1, 2, 3, 6, 7 of §14 by synthetic division.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page17

Exercise Page17, problem 1, p. 17
Find a cubic equation having the roots $0$, $1$, $2$.
Printed answer:
- $x^3 - 3x^2 + 2x = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page17, problem 2, p. 17
Find a quartic equation having the roots $±1$, $±2$.
Printed answer:
- $x^4 - 5x^2 + 4 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page17, problem 3, p. 17
Find a quartic equation having the two double roots $3$ and $-3$.
Printed answer:
- $x^4 - 18x^2 + 81 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page17, problem 4, p. 17
Find a quartic equation having the root $2$ and the triple root $1$.
Printed answer:
- $x^4 - 5x^3 + 9x^2 - 7x + 2 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page17, problem 5, p. 17
What is the condition that $ax^2+bx+c=0$ shall have a double root?
Printed answer:
- $b^2 = 4ac$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page17, problem 6, p. 17
If $a_0 x^n + \dotsb + a_n = 0$ has more than $n$ distinct roots, each coefficient is zero.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page17, problem 7, p. 17
Why is there a single answer to each of Exercises 1--4, if the coefficient of the highest power of the unknown be taken equal to unity? State and answer the corresponding general question.
Printed answer:
- By theorem in §18.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise Page19

Exercise Page19, problem 1, p. 19
Find a cubic equation having the roots $1$, $2$, $3$.
Printed answer:
- $x^3 - 6x^2 + 11x - 6 = 0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - 6*x**2 + 11*x - 6
Exercise Page19, problem 10, p. 19
Solve $x^4 - 2x^3 - 21x^2 + 22x + 40 = 0$, whose roots are in arithmetical progression. [Denote them by $c-3b$, $c-b$, $c+b$, $c+3b$, with the common difference $2b$]. % [** PP: Added period]
Printed answer:
- $5$, $2$, $-1$, $-4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5, 2, -1, -4]
Exercise Page19, problem 11, p. 19
Find a quadratic equation whose roots are the squares of the roots of $x^2-px+q = 0$.
Printed answer:
- $y^2 - (p^2 - 2q)y + q^2 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 12, p. 19
Find a quadratic equation whose roots are the cubes of the roots of $x^2 - px + q = 0$. Hint: $\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)$.
Printed answer:
- $y^2 - (p^3 - 3pq)y + q^3 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 13a, p. 19
If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i) $\alpha^2 / \beta$; and $\beta^2 / \alpha$; (ii) $\alpha^3\beta$ and $\alpha\beta^3$; (iii) $\alpha+1 / \beta$ and $\beta + 1 / \alpha$.
Printed answer:
- $y^2 - y(p^3 - 3pq)/q + q = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 13b, p. 19
If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i) $\alpha^2 / \beta$; and $\beta^2 / \alpha$; (ii) $\alpha^3\beta$ and $\alpha\beta^3$; (iii) $\alpha+1 / \beta$ and $\beta + 1 / \alpha$.
Printed answer:
- $y^2 - q(p^2 - 2q)y + q^4 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 13c, p. 19
If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i) $\alpha^2 / \beta$; and $\beta^2 / \alpha$; (ii) $\alpha^3\beta$ and $\alpha\beta^3$; (iii) $\alpha+1 / \beta$ and $\beta + 1 / \alpha$.
Printed answer:
- $y^2 - (p + p/q)y + 2 + q + 1/q = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 14, p. 19
Find a necessary and sufficient condition that the roots, taken in some order, of $x^3 + px^2 + qx + r = 0$ shall be in geometrical progression.
Printed answer:
- $p^3r = q^3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 15, p. 19
Solve $x^3 - 28x + 48 = 0$, given that two roots differ by $2$.
Printed answer:
- $2$, $4$, $-6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, 4, -6]
Exercise Page19, problem 2, p. 19
Find a quartic equation having the double roots $2$ and $-2$.
Printed answer:
- $x^4 - 8x^2 + 16 = 0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - 8*x**2 + 16
Exercise Page19, problem 3, p. 19
Solve $x^4 - 6x^3 + 13x^2 - 12x + 4 = 0$, which has two double roots.
Printed answer:
- $1$, $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, 2]
Exercise Page19, problem 4, p. 19
Prove that one root of $x^3 + px^2 + qx + r = 0$ is the negative of another root if and only if $r = pq$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page19, problem 5, p. 19
Solve $4x^3 - 16x^2 - 9x + 36 = 0$, given that one root is the negative of another.
Printed answer:
- $4$, $\tfrac{3}{2}$, $-\tfrac{3}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, Rational(3, 2), Rational(-3, 2)]
Exercise Page19, problem 6, p. 19
Solve $x^3 - 9x^2 + 23x - 15 = 0$, given that one root is the triple of another.
Printed answer:
- $1$, $3$, $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, 3, 5]
Exercise Page19, problem 7, p. 19
Solve $x^4 - 6x^3 + 12x^2 - 10x + 3 = 0$, which has a triple root.
Printed answer:
- $1$, $1$, $1$, $3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, 1, 1, 3]
Exercise Page19, problem 8, p. 19
Solve $x^3 - 14x^2 - 84x + 216 = 0$, whose roots are in geometrical progression, i.e., with a common ratio $r$ [say $m/r$, $m$, $mr$].
Printed answer:
- $2$, $-6$, $18$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, -6, 18]
Exercise Page19, problem 9, p. 19
Solve $x^3 - 3x^2 - 13x + 15 = 0$, whose roots are in arithmetical progression, i.e., with a common difference $d$ [say $m-d$, $m$, $m+d$].
Printed answer:
- $-3$, $1$, $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-3, 1, 5]

Exercise Page20

Exercise Page20, problem 1, p. 20
Solve $x^3 - 3x^2 - 6x - 20 = 0$, one root being $-1 + \sqrt{-3}$.
Printed answer:
- $5$, $-1±\sqrt{-3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5, -1 + sqrt(-3), -1 - sqrt(-3)]
Exercise Page20, problem 10, p. 20
Given that $x^4 - 2x^3 - 5x^2 - 6x + 2 = 0$ has the root $2 - \sqrt{3}$, find another root and by means of the sum and the product of the four roots deduce, without division, the quadratic equation satisfied by the remaining two roots.
Printed answer:
- $2 + \sqrt{3}$, $x^2 + 2x + 2 = 0$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2 + sqrt(3)]
Exercise Page20, problem 11, p. 20
Granted that a certain cubic equation has the root $2$ and no real root different from $2$, does it have two imaginary roots?
Printed answer:
- Not necessarily.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page20, problem 12, p. 20
Granted that a certain quartic equation has the roots $2 ± 3i$, and no imaginary roots different from them, does it have two real roots?
Printed answer:
- Not necessarily.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page20, problem 13, p. 20
By means of the proof of Ex. 5, may we conclude as at the end of §21 that every integral rational function with rational coefficients can be expressed as a product of linear and quadratic factors with rational coefficients?
Printed answer:
- No.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page20, problem 2, p. 20
Solve $x^4 - 4x^3 + 5x^2 - 2x - 2 = 0$, one root being $1-i$.
Printed answer:
- $1±i$, $1±\sqrt{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem [1 + I, 1 - I, 1 + sqrt(2), 1 - sqrt(2)]
Exercise Page20, problem 3, p. 20
Find a cubic equation with real coefficients two of whose roots are $1$ and $3+2i$.
Printed answer:
- $x^3 - 7x^2 + 19x - 13 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**3 - 7*x**2 + 19*x - 13
Exercise Page20, problem 4, p. 20
If a real cubic equation $x^3 - 6x^2 + \dotsb = 0$ has the root $1 +\sqrt{-5}$, what are the remaining roots? Find the complete equation.
Printed answer:
- $4$, $1-\sqrt{-5}$, $x^3 - 6x^2 + 14x - 24 = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page20, problem 5, p. 20
If an equation with *rational* coefficients has a root $a + \sqrt{b}$, where $a$ and $b$ are rational, but $\sqrt{b}$ is irrational, prove that it has the root $a - \sqrt{b}$. [Use the method of §21.]
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page20, problem 6, p. 20
Solve $x^4 - 4x^3 + 4x - 1 = 0$, one root being $2 + \sqrt{3}$.
Printed answer:
- $± 1$, $2±\sqrt{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, -1, 2 + sqrt(3), 2 - sqrt(3)]
Exercise Page20, problem 7, p. 20
Solve $x^3 - (4 + \sqrt{3})x^2 + (5 + 4\sqrt{3})x - 5\sqrt{3} = 0$, having the root $\sqrt{3}$.
Printed answer:
- $\sqrt{3}$, $2±i$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem [sqrt(3), 2 + I, 2 - I]
Exercise Page20, problem 8, p. 20
Solve the equation in Ex. 7, given that it has the root $2+i$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- solve: no printed answer to check
Exercise Page20, problem 9, p. 20
Find a cubic equation with rational coefficients having the roots $\frac{1}{2}, \frac{1}{2} + \sqrt{2}$.
Printed answer:
- $x^3 - \tfrac{3}{2}x^2 - \tfrac{5}{4}x + \tfrac{7}{8} = 0$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**3 - Rational(3, 2)*x**2 - Rational(5, 4)*x + Rational(7, 8)

Exercise Page23

Exercise Page23, problem 1, p. 23
$4x^5 - 8x^4 + 22x^3 + 98x^2 - 73x + 5 = 0$.
Printed answer:
- $19\tfrac{1}{4}$, $3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page23, problem 2, p. 23
$x^4 - 5x^3 + 7x^2 - 8x + 1 = 0$.
Printed answer:
- $6$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page23, problem 3, p. 23
$x^7 + 3x^6 - 4x^5 + 5x^4 - 6x^3 - 7x^2 - 8 = 0$.
Printed answer:
- $2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page23, problem 4, p. 23
$x^7 + 2x^5 + 4x^4 - 8x^2 - 32 = 0$.
Printed answer:
- $3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page23, problem 5, p. 23
A lower limit to the negative roots of $f(x) = 0$ may be found by applying our theorems to $f(-x) = 0$, i.e., to the equation derived from $f(x) = 0$ by replacing $x$ by $-x$. Find a lower limit to the negative roots in Exs. 2, 3, 4.
Printed answer:
- $0, -7, -\tfrac{7}{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page23, problem 6, p. 23
Prove that every real root of a real equation $f(x) = 0$ is less than $1 + g / a_0$ if $a_0 > 0$, where $g$ denotes the greatest of the numerical values of $a_1, \dotsc, a_n$. Hint: if $x>0$, a_0 x^n + a_1 x^n-1 + a_0 x^n - g(x^n-1 + + x + 1). Proceed as in §22 with $k = 1$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Page23, problem 7, p. 23
Prove that $1 + g \div |a_0|$ is an upper limit for the moduli of all complex roots of any equation $f(x)=0$ with complex coefficients, where $g$ is the greatest of the values $|a_1|, \dotsc, |a_n|$, and $|a|$ denotes the modulus of $a$. Hint: use Ex. 5 of §8.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles