Finding roots of polynomial equations. Rational integral equations. Remainder, factor theorems. Synthetic division. Variation of sign. Descartes’ Rule of Signs. Standard division formula. Depressed equation.

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Polynomial equations

Def. Polynomial equation. A polynomial in one or more variables, set equal to zero.

Syn. Rational integral equation

Examples.

3x⁴ + 5x² - 3x + 4 = 0

3x³ + 4x²y³ + xz² - 6xz + 3x + y - 8 = 0

5x² + 3xy + 7y² + 2x + 3y + 5 = 0

Degree of a polynomial equation. The degree of a polynomial equation is the degree of the associated polynomial (where the degree of the polynomial is the degree of the term of highest degree in the polynomial).

Examples. The degree of

3x³ + 4x²y³ + xz² - 6xz + 3x + y - 8 = 0

is the degree of the term 4x²y³, which is 5.

The degree of

5x² + 3xy + 7y² + 2x + 3y + 5 = 0

is 2.

Polynomial equations in one variable

Polynomial equations in one variable. The general form of a polynomial equation in one variable is

1) a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0

The term a₀xⁿ is called the leading term, a_n the constant term, and a₀ the leading coefficient.

If we divide 1) by a₀ we obtain

2) xⁿ + p₁x^n-1 + p₂x^n-2 + ... + p_n-1x + p_n = 0

which is referred to as the p-form of 1).

Standard form of a polynomial equation in one variable. A polynomial equation in one variable is said to be in standard form if it is written in the form of 1) above with the terms arranged in descending powers of x in which no term is missing, a zero coefficient being used on each missing term, and the leading coefficient a₀, if it is real, is positive.

Example. The standard form of the equation

4x² + 5 = 2x⁵

2x⁵ + 0∙x⁴ + 0∙x³ - 4x² - 5 = 0

The following fundamental theorem was proved by Gauss:

Fundamental Theorem of Algebra. Every polynomial equation

a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0

has at least one root, real or complex.

Standard division formula. Let r be any number. If Q(x) is the quotient obtained by dividing a polynomial G(x) by (x - r) and R is the constant remainder after the division, then, by definition,

3) G(x) (x - r)Q(x) + R

Remainder theorem. Let r be any number. If a polynomial G(x) is divided by (x - r) until a constant remainder is obtained, this remainder is equal to G(r) [i.e. equal to the number obtained by substituting r for x in the polynomial]. More concisely, for any polynomial G(x) and any number r,

4) G(x) (x - r)Q(x) + G(r)

Proof. Let a polynomial G(x) be divided by x - r until a constant term is obtained. Then by 3)

5) G(x) (x - r)Q(x) + R

This is an identity, true for all values of x. If we substitute x = r into 5) we get

G(r) = (r - r)Q(x) + R = 0 ∙ Q(x) + R = R

R = G(r).

Example. If G(x) = x⁴ - 2x³ + 3x² - x + 2 is divided by x - 3, the quotient is Q(x) = x³ + x² + 6x + 17 and the remainder is 53. If we substitute r = 3 into G(x) we get 3⁴ - 2 ∙ 3³ + 3 ∙ 3² - 3 + 2 = 53.

Factor Theorem. If r is a root of a polynomial equation

G(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0

then (x - r) is a factor of G(x), i.e.

G(x) = (x - r)Q(x)

where Q(x) is a polynomial of degree one less than that of G(x).

Conversely, if (x - r) is a factor of G(x), then r is a root of G(x) = 0.

Proof. By the Remainder Theorem,

G(x) = (x - r)Q(x) + G(r) .

If x - r is a factor of G(x), the remainder must be zero when G(x) is divided by x - r. Thus, G(r) = 0.

Conversely, if r is a root of G(x), then by definition, G(r) = 0. If G(r) = 0, then by the Remainder Theorem, G(x) = (x - r)Q(x) and x - r is a factor of G(x).

The following theorem follows directly by repeated application of the Fundamental Theorem of Algebra and the Factor Theorem.

Theorem 1. Any polynomial of degree n

G(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n

can be written as the product of n linear factors:

G(x) = C (x - r₁)(x - r₂) .... (x - r_n)

where C = a₀. These factors need not all be distinct (i.e. some of the r’s may be equal). Some of the r’s may be real numbers while others may be complex numbers.

Theorem 2. Imaginary roots. If a polynomial equation G(x) = 0 has real coefficients and if the imaginary number a + bi is a root of G(x) = 0, then the conjugate imaginary a - bi number is also a root. In other words, imaginary roots occur only in pairs, if the coefficients of the equation are real.

Proof

It follows from Theorem 2 that if a + bi is a root, then G(x) contains the quadratic factor (x² - 2ax + a² + b²), corresponding to the product [x - (a + bi)][x - (a - bi)].

Theorem 3. Irrational roots. If a polynomial equation G(x) = 0 has rational coefficients and if the irrational number a + , where a and b are rational, is a root of G(x) = 0, then the conjugate irrational a - is also a root.

Theorem 4. Any polynomial

G(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n

with real coefficients can be factored into a product of linear and irreducible quadratic factors with real coefficients:

G(x) = C(x - r₁)(x - r₂) .... (x² + p₁x + q₁)(x² + p₂x + q₂) .....

Roots of a polynomial equation. Every polynomial equation G(x) = 0 of degree n has exactly n roots. Some may be equal. The roots may be either real or complex numbers. Complex roots occur in pairs.

Example. The polynomial of the sixth degree

(x - 3)² (x - 7)³ (x + 2) = 0

has 3 as a double root, 7 as a triple root and -2 as a single root i.e. the six roots are 3, 3, 7, 7, 7, -2.

Theorem 5. If two polynomials of degree n in the same variable x are equal for more than n values of x, the coefficients of like powers are equal and the two polynomials are identically equal.

Relation between roots and coefficients. Viete’s formulas. In the equation in the p-form, in which the coefficient of the first term is 1,

xⁿ + p₁x^n-1 + p₂x^n-2 + ... + p_n-1x + p_n = 0

the following relations exist between coefficients and roots:

1) -p₁ = sum of roots

2) p₂ = sum of products of roots taken two at a time

3) -p₃ = sum of products of roots taken three at a time

4) (-1)ⁱp_i = sum of products of roots taken i at a time

5) (-1)ⁿp_n= product of all of the roots

This relationship between the coefficients of the polynomial and its roots was discovered by Francois Viete and the formulas for the coefficients are known as Viete’s formulas. See Elementary symmetric polynomials. Viete’s Formulas.

Synthetic division. Synthetic division is a simplified method of dividing a polynomial G(x) by x - r, where r is any number. It is an abbreviated version of the usual method of long division of polynomials, reducing time and labor dramatically.

To divide the polynomial

G(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n

by x - r, write

a₀ a₁ a₂ a₃ ..... a_n | r

a₀r b₁r b₂r ..... b_n-1r

----------------------------------------------------

a₀ b₁ b₂ b₃ ..... R

where each number below the line is the sum of those above. Then

Q(x) = a₀x^n-1 + b₁x^n-2 + b₂x^n-3 + ....

where

G(x) = (x - r)Q(x) + R

G(x) above is in standard form with missing terms appearing with zeros as coefficients. The numbers in the second and third lines are computed in the following sequence:

a₀, a₀r, b₁, b₁r, b_2,b₂r, b₃, .... , b_n-1r, R

Example. Divide 5x⁴ - 8x² - 15x - 6 by x - 2 using synthetic division.

5 +0 -8 -15 -6 | 2

10 +20 +24 +18

-----------------------------------------

5 +10 +12 +9 +12

Quotient: Q(x) = 5x³ + 10x² + 12x + 9

Remainder: R = 12

Transformation of equations

Theorem. If we replace x by -x in the equation G(x) = 0, the resulting equation G(-x) = 0 has roots that are the negatives of the roots of the given equation.

Changing the sign of each root. In order to obtain an equation whose roots are the negatives of the roots of a given equation, reverse the signs of the terms of odd degree; alternatively, reverse the signs of the terms of even degree, leaving the terms of odd degree unchanged.

Finding the roots of polynomial equations in one variable

In seeking the roots of a polynomial G(x) = 0 we seek those values of x at which the graph of the polynomial crosses the x axis. Thus we generally want to know what the graph of the polynomial looks like. So we plot the polynomial. We do this by computing the value of the polynomial at various points along the x axis, creating a table of values of G(x) as a function of x. We can compute these values of G(x) by plugging different values of x into the polynomial and evaluating it but there is another way of computing the values that is generally far faster and easier. It is done by the use of the Remainder Theorem in conjunction with synthetic division. According to the Remainder Theorem, if a polynomial G(x) is divided by (x - r) until a constant remainder is obtained, this remainder is equal to G(r). So we compute G(r) by

dividing G(x) by (x - r) with synthetic division. This provides a way of computing the table that is much faster.

There are a number of theorems that provide information on the graphs of polynomials and on the number, nature and location of the roots. We list some:

Theorem 1. The graph of a polynomial y = G(x) is a smooth continuous curve i.e. a curve with no breaks and no sharp corners.

Theorem 2. The value of a polynomial y = G(x) at the point x = 0 is equal to its constant term.

Theorem 3. If G(x) is a polynomial of degree n in which the coefficient of xⁿ is positive, G(x) is positive for very large values of x; for numerically large negative values of x, G(x) is positive if n is even and negative if n is odd.

The above theorem follows from the fact that when x is sufficiently large numerically, the term in xⁿ is numerically larger than all the other terms taken together and, as a consequence, its sign determines the sign of the entire polynomial.

Graph of a factored polynomial. We now consider the effect of multiple roots (replicated factors) on a graph.

Theorem 4. Let a polynomial G(x) contain the factor (x - a) exactly k times, where a is real. We have three cases:

k = 1. The graph of G(x) crosses the x axis at x = a and is not tangent to it there.

k is odd and k > 1. The graph of G(x) crosses the x axis at x = a and is tangent to it there.

k is even. The graph of G(x) is tangent to the x axis at x = a but not cross it there.

See Fig. 1. In Fig. 2 is shown a plot of

y = (x + 1)²(x - 2)³(x - 4)

At x = 4, the graph crosses the x axis without being tangent to it. At x = 2, the graph crosses the x axis and is tangent to it. At x = -1, the graph is tangent to the x axis, but does not cross it.

For some insight into the theorem consider the following:

If x < a, then (x - a) < 0 and if x > a, (x - a) > 0. Consequently, if x increases from a value that is less than a to a value that is greater than a, the quantities (x - a), (x -a)³, (x - a)⁵, ... change sign from negative to positive; the even powers of (x - a), such as (x - a)² and (x - a)⁴ become equal to 0 at x = a but do not change sign there.

The Location Theorem. If a and b are real numbers and if G(a) and G(b) have opposite signs, the equation G(x) = 0 has at least one root between x = a and x = b.

This theorem follows from the fact that the function is continuous. It is useful in isolating roots.

Def. Variation of sign. If the terms of a polynomial are arranged in order of descending powers of the variable, a variation of sign is said to occur when two consecutive terms differ in sign.

Examples. 5x³ - 3x² + 4x -5 has three variations of sign

3 x⁶ + 2x³ - 3x² + 4x + 2 has two variations of sign

Descartes’ Rule of Signs. Let G(x) = 0 be a polynomial equation G(x) = 0 with real coefficients. Then

1. The number of positive roots is either equal to the number of variations of sign of G(x) or is less than that number by an even integer.

2. The number of negative roots is either equal to the number of variations of sign of G(-x) or is less than that number by an even integer.

Theorem 5. If G(x) has one sign when x = 0 and the opposite sign for numerically large positive (or negative) values of x, G(x) has at least one positive (or negative) root.

Upper and lower limits for the roots. A real number L is called an upper limit of the real roots of G(x) = 0 if no root is greater than L. A real number l is called an lower limit of the real roots of G(x) = 0 if no root is less than l.

Theorem 6.

1. If α > 0 and if, when a polynomial G(x) is divided by x - α by synthetic division, every number in the third line is positive or zero, then α is an upper limit of the real roots of G(x) = 0.

2. If α < 0 and if, when a polynomial G(x) is divided by x - α by synthetic division, the numbers in the third line alternate in sign, then α is a lower limit of the real roots of G(x) = 0.

Theorem 7. If α is an upper limit of the roots of G(-x) = 0, then l = -α is a lower limit of the roots of G(x) = 0.

Diminishing each root by a constant. To obtain an equation, each of whose roots is h less than the corresponding root of the equation

G(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0 ,

compute the following R₁, R₂, .... , R_n , employing synthetic division.

G(x) = (x - h)∙Q₁(x) + R₁

Q₁(x) = (x - h)∙Q₂(x) + R₂

.........................................

........................................

Q_n-2(x) = (x - h)∙Q_n-1(x) + R_n-1

Q_n-1(x) = (x - h)∙Q_n(x) + R_n

where each R is a constant and Q_n(x) = a₀. Then the roots of

g(y) = a₀yⁿ + R_ny^n-1 + R_n-1y^n-2 + ... + R₂y + R₁ = 0

are the roots of G(x) diminished by h.

Proof

Def. Depressed equation. The equation resulting from reducing the number of roots in an equation. For example, one root of G(x) = x³ - 3x² + 4x - 2 = 0 is 1. Thus x - 1 is a factor of G(x). The depressed equation obtained from G(x) = 0 by dividing G(x) by x - 1 is x² - 2x + 2 = 0.

Problem. One root of x³ + 2x² - 23 x - 60 = 0 is 5. Find the depressed equation.

Solution. Divide x³ + 2x² - 23 x - 60 by x - 5 using synthetic division.

1 +2 -23 -60 | 5

+5 +35 +60

-------------------------------

1 +7 +12 +0

The depressed equation is x² + 7x + 12 = 0

Finding rational roots of a polynomial

Theorem 8. Let p/q be a rational fraction in lowest terms and

a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0

be a polynomial equation with integral coefficients. If p/q is a root of the equation, then p is a factor of a_n and q is a factor of a₀.

Proof

Problem. Find the rational roots of the equation

2x³ + x² - 7x - 6 = 0

Solution. If a rational number p/q is a root, then p is limited to the factors of 6 which are

±1, ±2, ±3, ±6 and q is limited to the factors of 2 which are ±1, ±2. Consequently, the list of all possible rational roots is: ±1, ±2, ±3, ±6, ±1, ±2, ±3, ±6, ±½, ±3/2. Testing each of these values, we obtain -1, 2, -3/2 as the rational roots.

It follows from the above theorem that if a polynomial equation G(x) = 0 with integral coefficients is in the p-form (in which the coefficient of the highest power of x is 1)

xⁿ + p₁x^n-1 + p₂x^n-2 + ... + p_n-1x + p_n = 0

then any rational root of G(x) = 0 is an integer and a factor of p_n.

If the given equation is in the form

a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0

it is often best not to try to find the roots directly but rather to transform the equation by a change of variables, substituting x = y/a₀. If one then clears the equation of fractions by multiplying both sides by a₀ ^n-1, the new equation will be in the p-form and any rational root of that equation will be an integer.

Problem. Find the rational roots of the equation

3x³ - 8x² + x + 2 = 0

Solution. Let x = y/3. Then

y³ - 8y² + 3y + 18 = 0

The factors of 18 are ±1, ±2, ±3, ±6, ±9, ±18. Substituting y = 2 makes the left member zero and so y = 2 is a root. Thus y - 2 is a factor. Using synthetic division we obtain the depressed equation.

1 -8 3 18 | 2

2 -12 -18

------------------------------

1 -6 -9 0

Thus the depressed equation is y² - 6y - 9 = 0.

The quadratic formula gives its roots as

Substituting back , using x = y/3, we obtain the roots of the original equation as x = 2/3 and

Procedure for plotting the graph of a polynomial. We plot a polynomial by computing the value of the polynomial at various points along the x axis, creating a table of values of G(x) as a function of x. A convenient procedure is as follows:

1. When x = 0, y = G(0) is the constant term of the polynomial.

2. Use synthetic division to find G(1), G(2), G(3), .... stopping as soon as the numbers in the third line of the synthetic division have the same sign.

3. Use synthetic division to find G(-1), G(-2), G(-3), .... stopping as soon as the numbers in the third line of the synthetic division have alternating signs.

Procedure for finding the real roots of a polynomial. The procedure for finding the real roots of a polynomial is to first find the approximate values of the roots by plotting the function and then to employ one of a number of available iterative computational techniques to home in on the root through a succession of repeated approximations. The repeated approximations take you closer and closer to the true value, allowing you to compute the root to any desired accuracy.

References

Hawks, Luby, Touton. Second-Year Algebra

Murray R. Spiegel. College Algebra (Schaum)

Raymond W. Brink. A First Year of College Mathematics

Frank Ayres. First Year College Mathematics (Schaum)

James / James. Mathematics Dictionary

The International Dictionary of Applied Mathematics. D. Van Nostrand Co.