# Synthetic Substitution Examples and Polynomial Theorems

Consider :

What is

What is

Note that

If we substitute either x=-3 or x=2 into the parenthesized
expression we can see that the numbers

that appear in the 2nd and 3rd rows of the Synthetic Substitution process really
come out of these direct

substitutions and calculations. It is for this reason that we prefer to use the
term Synthetic Substitution

as opposed to Synthetic Division whenever we calculate and fill in a table like
those above. In fact, it is

most instructive to directly substitute B instead of a number to fill in a
Synthetic Substitution table like

the one shown below. Counting the number of B's that follow each coefficient in
parentheses in the last

entry shows that the final polynomial is the same as

Now consider the long division of p(x) by x + 3.

Note the coefficients in the quotient polynomial are the
same as the first numbers in the last row of the

first table above. Also note that the remainder is -11 and this is the last
number in the third row of the

first table above. These are reasons why some people use the term Synthetic
Division

Next consider the polynomial that is the product of the following 4 linear factors:

Note that the product of the four roots is:

The sum of the four roots is:

The next computations illustrate the analysis of **
Upper and Lower bounds.**

Next, consider the polynomial

We can also substitute with **irrational numbers.**

Note that the product of the three roots is:

The sum of the roots is

Next, consider the polynomial

We can also substitute with **complex numbers.**

Note that the product of the three roots is:

The sum of the three roots is:

The following 31 items summarize some of the most
important theorems about polynomials. See the

note at the end of this list if you are interested in reading any of the
detailed proofs of these theorems.

**1. The Division Algorithm
**If are any two polynomials then there
exist unique polynomials q(x) and r(x)

such that where the degree of r(x) is stricly less than the degree of d(x)

when the degree of or else

**2. The Division Check for a Linear Divisor
**Consider dividing the polynomial p(x) by the
linear term (x-a) Then, the

*states*

**Division Check**that:

**3. Remainder Theorem
**When any polynomial p(x) is divided by (x-a) the remainder is
p(a)

**4. Factor Theorem
**(x-a) is a factor of the polynomial
p(x) if and only if p(a) = 0

**5. Maximum Number of Zeros Theorem
**A polynomial cannot have more real zeros than its degree.

**6. Fundamental Theorem of Algebra
**a) Every polynomial of degree has at
least one zero among the complex numbers.

b) If p(x) denotes a polynomial of degree 8ß then p(x) has exactly 8 roots, some of

which may be either irrational numbers or complex numbers.

**7. Product and Sum of the Roots Theorem
**Let be any polynomial with real

coefficients with a

**leading coefficient of***where . Then is times the product of all*

**1**the roots of p(x) = 0 and is the opposite of the sum of all the roots of p(x) = 0

8. Rational Roots Theorem

Let be any polynomial

with integer coefficients. If the reduced rational number
is a root of
p(x) = 0 then must be a -

factor of a_{0} and d must be a factor of a_{n}

**9. Integer Roots Theorem
**Let : be any polynomial

with integer coefficients and

*. If p(x) has any rational zeros,*

**with a leading coefficient of 1**then those zeros must all be integers.

**10. Upper and Lower Bounds Theorem
**Let p(x) be any polynomial with

*and*

**real coefficients**

**a positive leading coefficient.**(

*) If and and if in applying synthetic substitution to computep(a) all*

**Upper Bound**numbers in the 3rd row are positive, then + is an upper bound for all the roots of p(x) = 0

(

*If and and if in applying synthetic substitution to compute p(a) all*

**Lower Bound)**the numbers in the 3rd row alternate in sign then + is a lower bound for all the roots of p(x) = 0

[ In either bound case, we can allow any number of zeros in any positions in the third row except in

the first and last positions. The first number is assumed to be positive and the last number is

For upper bounds, we can state alternatively and more precisely that no negatives are

allowed in the 3rd row. In the lower bound case the alternating sign requirement is not strict either,

as any 0 value can assume either sign as required. In practice you may rarely see any zeros in the

3rd row. However, a slightly stronger and more precise statement is that the bounds still hold even

when zeros are present anywhere as interior entries in the 3rd row.]

**11. Intermediate Value Theorem
**If p(x) is any polynomial with

*and if then*

**real coefficients,**there is at least one real number - between + and , such that p(c) = 0

**12. Single Bound Theorem
**Let be any polynomial with

real coefficients and a leading coefficient of 1. Let and

let . Finally let Then every

zero of p(x) lies between -M and M.

**13. Odd Degree Real Root Theorem
**If p(x) has real coefficients and has a
degree that is odd then it has at least one real root.

**14. Complex Conjugate Roots Theorem
**If p(x) is any polynomial with

*and if , is a complex root of*

**real coefficients,**the equationp(x) = 0 then another complex root is its conjugate

(Complex number roots appear in conjugate pairs)

**15. Linear and Irreducible Quadratic Factors Theorem
**Let p(x) be any polynomial with real
coefficients. Then p(x) may be written as a
product of linear

factors and irreducible quadratic factors. The sum of all the degrees of these component factors is

the degree of p(x)

**16. Irrational Conjugate Roots Theorem
**Let p(x) be any polynomial with
rational real coefficients. If is a root of
the

equation p(x) = 0 where is irrational and + and , are rational, then another root is

(Like complex roots, irrational real roots appear in conjugate pairs, but only when the polynomial

has rational coefficients.)

**17. Descartes's Rule of Signs Lemma 1.
**If p(x) has real coefficients, and if
p(a) = 0 wherea > 0
thenp(x) has at least one more sign

variation than the quotient polynomial q(x) has sign variations where

[When the difference in the number of sign variations is greater than 1, the difference is always

an odd number.]

**18. Descartes's Rule of Signs Lemma 2.
**If p(x) has real coefficients, the number of
positive zeros of p(x) is not greater than the

number of variations in sign of the coefficients of p(x)

**19. Descartes's Rule of Signs Lemma 3.
**Let denote 5 positive numbers and let

Then the coefficients of p(x) are all alternating in sign and this polynomial has exactly 5 sign

variations in its coefficients.

**20. Descartes's Rule of Signs Lemma 4.
**The number of variations in sign of a polynomial with real coefficients is
even if the first and last

coefficients have the same sign, and is odd if the first and last coefficients have opposite signs.

**21. Descartes's Rule of Signs Lemma 5.
**If the number of positive zeros of p(x) with
real coefficients is less than the number of sign

variations in p(x) it is less by an even number.

**22. Descartes's Rule of Signs Lemma 6.
**Each negative root of p(x) corresponds to a
positive root of p(-x) That is, if

and + is a zero of p(x) then -a is a positive zero of p(-x)

**23. Descartes's Rule of Signs
**Let p(x) be any polynomial with

**real coefficients.**(

*) The number of positive roots of p(x) = 0 is either equal to the*

**Positive Roots**number of sign variations in the coefficients of p(x) or else is less than this

number by an even integer.

(

*) The number of negative roots of p(x) = 0 is either equal to the*

**Negative Roots**number of sign variations in the coefficients of p(-x) or else is less than

this number by an even integer.

Note that when determining sign variations we can ignore terms with zero coefficients.

**24. Lemma On Continuous Functions.
**Letf(x) and
g(x) be two continuous real-valued functions with
a common domain that is

an open interval (a,b), Furthermore let and assume that except when x = c we have

for all Then we must also have

**25. Theorem On the Equality of Polynomials
**Let and let

be any two real polynomials of

degrees n and m respectively. If for all real numbers B, p(x) = q(x) then

1) m=n

and 2) for all i, if

**26. Theorem Euclidean Algorithm for Polynomials
**Let p(x) and
q(x) be any two polynomials with degrees
≥1. Then there exists a polynomial
d(x)

such that d(x) divides evenly into both p(x) and q(x). Moreover, d(x) is such that if a(x) is any

other common divisor of p(x) and q(x), then a(x) divides evenly into d(x). The polynomial d(x) is

called the Greatest Common Divisor of p(x) and q(x) is sometimes denoted by GCD(p(x),q(x))

Except for constant multiples , d(x) is unique.

**27. Corollary to the Euclidean Algorithm for
Polynomials
**The of any two polynomials
p(x) and q(x)
may be expressed as a linear combination of p (x)

and q(x)

**28. Lemma 1 for Partial Fractions
**If where
then there exist polynomials
d(x) and e(x)
such

that

**29. Lemma 2 for Partial Fractions
**If then there exists a polynomial
g(x) and for
there exist polynomials

each with degree less than q(x) such that

**30. Partial Fraction Decomposition Theorem
**Let be a rational function where
p(x) and q(x)
are polynomials such that the degree of p(x)

is less than the degree of q(x) Then there exist algebraic fractions such that

and where each fraction is one of two forms:

where , , are all real numbers and

the and the are positive integers and each quadratic expression has a

negative

discriminant.

**31. Partial Fraction Decomposition Coefficient Theorem
**Let be a rational function where
p(x) and q(x) are polynomials such that the degree of
p(x)

is less than the degree of q(x). If x = a is a root of q(x) = 0 of multiplicity ", then in the partial

fraction decomposition of which contains a term of the form , the constant

Detailed proofs of all the above theorems may be found on
the author's web site:

*homepage.smc.edu\kennedy_john*

in a 31-page paper titled Some Polynomial Theorems. See the section on the
author's web site that is

titled Downloadable Papers.

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