Unit 5 : Polynomials
Monomial
An expression that is either a numeral, a variable, or the product of a numeral and
one or more variables.A monomial in x is a single term of the form axn, where a is a
real number and n is a whole number.
The following are monomials in x:
−6.3x 2
5x3
We may say that the number 2 is a monomial in x, because 2 = 2x0
Terms
When numbers or monomials are added or subtracted, they are called terms.
4x2 + 7x − 8
is a sum of three terms. (In algebra we speak of a "sum," even though a term may be
subtracted.)
The degree of a term is the sum of the exponents of all the variables in that term.
In functions of a single variable, such as x, the degree of a term is simply the
exponent.
Example 1. The term 5x3 is of degree 3 in the variable x.
Example 2. This term 2xy2z3 is of degree 1 + 2 + 3 = 6 in the variables x, y, and z.
Example 3. Here are all possible terms of the 4th degree in the variables x and y:
x4, x3y, x2y2, xy3, y4.
In each term, the sum of the exponents is 4. As the exponent of x decreases, the
exponent of y increases.
Polynomial
in x is a sum of monomials in x.
Example 1. 5x3 − 4x2 + 7x − 8.
The variable, in this case x, is also called the argument of the polynomial
The leading term of a polynomial is the term of highest degree.
Example 5. The leading term of this polynomial 5x3 − 4x2 + 7x − 8 is 5x3.
The leading coefficient of a polynomial is the coefficient of the leading term.
Example 6. The leading coefficient of that polynomial is 5.
The degree of a polynomial is the degree of the leading term.
Example 7. The degree of this polynomial 5x3 − 4x2 + 7x − 8 is 3.
Here is a polynomial of the first degree: x − 2.
1 is the highest exponent.
The constant term of a polynomial is the term of degree 0; it is the term in which the
variable does not appear.
Example 8. The constant term of this polynomial 5x3 − 4x2 + 7x − 8 is −8.
The constant term of this polynomial -ax3 + bx2 + cx + d
-- is d.
like terms are those with identical letters and powers
-
Operations with polynomials have already studied ( sum, substract , multiply)
Algebraic Identities
Square of a Binomial
(a ± b)2 = a2 ± 2 · a · b + b2
(x + 3)2 = x 2 + 2 · x ·3 + 32 = x 2 + 6 x + 9
(2x − 3)2 = (2x)2 − 2 · 2x · 3 + 32 = 4x2 − 12 x + 9
Difference of Squares
(a + b) · (a − b) = a2− b2
(2x + 5) · (2x - 5) = (2x)2− 52 = 4x2 − 25
Divide
To divide two polynomials the degree of the dividend has to be greater or equal than
the degree of
the divisor.
If P(x) is the dividend, Q(x) is the divisor, C(x) is the quotient and R(x) is the
remainder:
P(x)
Q(x)
C(x)
R(x)
P(x)=Q(x)·C(x)+R(x)
The degree of the remainder is always less than the degree of the divisor.
Ruffini's Rule (Synthetic division)
Synthetic division (Ruffini's Rule) is a shorthand method of polynomial division in
the special case of dividing by a linear factor x−a , and it only works in this case.
Synthetic division is also useto find zeroes or roots of the polynomial.
In mathematics, Ruffini's Rule allow us the rapid division of a polynomial P(x) by a
polynomial like x−a .
Example:
In order to explain the steps to implement Ruffini's rule, an example division will be
used throughout the explanation:
(x4 − 3x2 + 2 ) : (x − 3)
1 If the polynomial is not complete, complete it by adding the missing terms with
zeros.
2Set the coefficients of the dividend in one line.
3In the bottom left, place the opposite of the independent term of the divisor.
4 Draw a line and lower the first coefficient.
5 Multiply this coefficient by the divisor and place it under the following term.
6 Add the two coefficients.
7 Repeat the process above.
Repeat the process:
Repeat, again:
8The last number obtained, 56, is the remainder.
9The quotient is a polynomial of lower degree and whose coefficients are the ones
obtained in the division.
x3 + 3 x2 + 6x +18
Evaluating Polynomials
Evaluating a polynomial is to find its numerical value when the variable x
is replaced by any number.
P(x) = 2x3 + 5x − 3 ; x = 1
P(1) = 2 · 13 + 5 · 1 − 3 = 2 + 5 - 3 = 4
Remainder Theorem
The remainder of the division of a polynomial P(x) and a polynomial of
the form x − a is the numerical value of this polynomial for the value: x = a.
Example
Calculate the remainder of the division P(x) : Q(x)
P(x)= x4 − 3x2 + 2
Q(x) = x – 3
P(3) = 34 − 3 · 32 + 2 = 81 − 27 + 2 = 56
Factor Theorem
The polynomial P(x) is divisible by a polynomial of the form (x − a) if
and only if P(a) = 0.
The value x = a is called the root or zero of P(x).
Roots of a Polynomial
These are the values to nullify the polynomial.
Calculate the Roots of the Polynomial:
P(x) = x2− 5x + 6
P(2) = 22 − 5 · 2 + 6 = 4 − 10 + 6 = 0
P(3) = 32 − 5 · 3 + 6 = 9 − 15 + 6 = 0
x = 2 and x = 3 are roots or zeros of the polynomial: P(x) = x2 − 5x + 6,
because P(2) = 0 and P(3) = 0.