Synthetic Substitution Examples and Polynomial Theorems

Consider :

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 Division Check states
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 1 where . Then is times the product of all
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 a₀ and d must be a factor of a_n

9. Integer Roots Theorem
Let : be any polynomial
with integer coefficients and with a leading coefficient of 1. If p(x) has any rational zeros,
then those zeros must all be integers.

10. Upper and Lower Bounds Theorem
Let p(x) be any polynomial with real coefficients and a positive leading coefficient.
(Upper Bound) If and and if in applying synthetic substitution to computep(a) all
numbers in the 3rd row are positive, then + is an upper bound for all the roots of p(x) = 0
(Lower Bound) If and and if in applying synthetic substitution to compute p(a) all
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 real coefficients, and if then
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 real coefficients, and if , is a complex root of
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.
(Positive Roots) The number of positive roots of p(x) = 0 is either equal to the
number of sign variations in the coefficients of p(x) or else is less than this
number by an even integer.
(Negative Roots) The number of negative roots of p(x) = 0 is either equal to the
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.