Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS.
Polynomials:
A polynomial is an algebraic expression that consists of a sum of several monomials.
Remember that a monomial is an algebraic expression as ax n , where a is a real number, and n is
a non negative integer.
The standard form of a polynomial is :
P  x =a n x na n−1 x n−1...a 1 xa 0
Here, n denotes the highest power to which x is raised; this highest exponent is called the degree of
the polynomial. Thus, in standard form, the highest power term is listed first, and subsequent
powers are listed in decreasing order.
The monomial a n x n , which is the monomial with the highest exponent of the variable, is called
the leading term. The number a 0 , which is the term with the exponent zero of the variable, is
called the constant term.
For instance, the algebraic expression 3x 5−x 34x 2−7x4 is a polynomial:
•
It has five terms: 3x 5 , −x 3 , 4x 2 , −7x and 4.
•
The degree is 5, since this is the highest exponent of the variable x. You can say: it is a fifthdegree polynomial.
•
The leading term is 3x 5 , while the constant term is 4.
Numerical value of a polynomial:
“Evaluating” a polynomial is the same as calculating its numerical value at a given value of the
variable: you plug in the given value of x, and figure out what the polynomial is supposed to be.
Example: What are the numerical values of the polynomial
x=2 and x=−1 ?
P 2
P −1
1
P  x =x 3−2x2 3x−5 at the values
Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Adding and subtracting polynomials:
Only like terms (those with identical letters and powers) can be added or subtracted.
3xy and −5xy are like terms.
x 3 and 7 x 2 are unlike terms, because the powers of x are not the same.
Addition or subtraction of polynomials are achieved by adding or subtracting like terms.
Examples:
a) Given the polynomials: P  x =x 45x3− x−6 and Q x=x 3−4x 23x−2 , calculate
P  x Q x  and P  x −Q x .
b) Given the polynomials: P  x =x 4 −4x 32x2 −3x1 and Q x= x 3−5x 2x−3 , calculate
P  x Q x  and P  x −Q x .
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Multiplying polynomials:
The product of two polynomials is calculated by the multiplication of all monomials of the two
polynomials.
Example: Given the polynomials:
P  x · Q x  .
P  x =x 2−4x1 and Q x= x 2−2x3 , calculate
Dividing polynomials:
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)
R(x)
C(x)
P(x)=Q(x)·C(x)+R(x)
The degree of the remainder is always less than the degree of the divisor.
Example: Calculate the quotient and remainder of the divisions:
a)  x 4−3x 34x 2−2x−5: x 2−2x3
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
b)  x 3−5x7: x 2−3
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 use
to find zeroes or roots of the polynomial.
In mathematics, Ruffini's Rule allows us the rapid division of a polynomial
polynomial like x−a . The process is shown with the example below:
P  x  by a
Example:
If we want to work out the division : 3x3 2x 2−5: x1 .
P  x =3x 32x 2−5 is the dividend
Q x=x 1 is the divisor.
The main problem, we first find, is that Q x is not a binomial of the form
rewrite it in this way:
Q x= x−−1
Now, we are going to apply the algorithm:
1. Write down the coefficients and a. Note that,
3
as P  x  doesn't contain a coefficient for x, we -1
write 0:
2
0
-5
2. Pass the first coefficient down:
2
0
-5
3
-1
3
4
x−a . We must
Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
3. Multiply the last value by a:
3
-1
2
-3
0
-5
2
-3
-1
0
-5
3
4. Add the values:
3
-1
3
5. Repeat steps 3 and 4 until we finish:
3
2
0
-5
-1
-3 1
-1
3 -1
1
-6 (remainder)
(result coefficients)
So, the division 3x3 2x 2−5: x1 has a quotient C  x =3x 2−x 1 and a remainder
R x=−6 .
Realize that the quotient is a polynomial of lower degree (one unit less of the degree of the
dividend) and the remainder is always a constant term.
Examples: Calculate the quotient and remainder of the following divisions, using Ruffini's Rule:
a)  x 3−5x 2x 10: x−2
b)  x 4−3x 34x−6: x2
c)  x 4− x 3x 2−2x2: x1
d) 2x 3x−1: x−3
e)  x 3−1:  x−1
f)  x 4−3x 2 7: x2
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Remainder Theorem:
The remainder of the division
when x=a , P a  .
P  x :  x−a  is the numerical value of the polynomial
Px
Examples:
1. Calculate the remainder of the division:  x 3−2x 23x−4: x−2 :
a) Calculating the division by Ruffini's Rule:
b) Using the Remainder Theorem:
2. Calculate the numerical value of the polynomial
P  x =x 4 x 2−3x6 when
x=−1 .
a) Using the definition of numerical value:
b) Applying the Remainder Theorem:
Factor Theorem: If P(a)=0 then x-a is a factor or a divisor of the polynomial P(x).
Example:
a) Calculate the numerical value of the polynomial
P  x =x 3−3x 2 5x−6 when
b) Calculate the division:  x 3−3x 25x−6 : x−2 .
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x=2 .
Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Roots of a polynomial: A real number a is a root or a zero of a polynomial P(x) if P(a)=0.
Properties:
•
If an integer a is a root of a polynomial P(x), this number a will be a factor or divisor of the
constant term of P(x).
•
The number of roots of a polynomial is always less or equal than the degree of the
polynomial.
Factoring Polynomials:
Factoring a polynomial is the opposite process of multiplying polynomial. Recall that when we
factor a number, we are looking for prime numbers that multiplying together to give the number, for
example: 6=2 ·3 , 12=22 ·3 .
When we factor a polynomial, we are looking for simpler polynomial that can be multiplied
together to give us the polynomial we started with.
Factoring a polynomial is to write it as a product of polynomials with the lowest possible degree.
Factoring polynomials can be done by:
– Common Factors.
– Special Products.
– Ruffini's Rule.
Examples: Factorise the following polynomials:
a)
x 3−7x6
b)
7
x 3−6x 211x−6
Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
c) x 3−3x 22x
d)
x 4 −1
b)
x 4 −6x3 4x 26x−5
Your
Turn
1. Factorise the following polynomials:
a)
x 32x 2−x−2
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
c)
x 4 −3x3 4x
d)
x 4 −2x 3−3x 24x4
e)
x 3−2x 2x−2
f)
x 4 −x
g)
x −5x 6x
h)
x −4x
4
3
2
9
4
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Algebraic Fractions:
An algebraic fraction (or a rational expression) is a fraction whose numerator and denominator are
polynomials.
Two algebraic fractions
Px
and
Q x
R x 
S  x
are equivalent if
P  x · S  x =Q x · R  x .
Algebraic fractions behave the same as numerical fractions. So we can simplify, add, subtract,
multiply or divide them, using the same rules.
Simplifying algebraic fractions:
You can simplify algebraic fractions by cancelling common factors in numerator and denominator
to reach an equivalent fraction.
Examples: Simplify:
a)
x 22x
x 24x4
b)
5x−5
5x−10
c)
x −3
x 2−9
d)
x 2 −1
x 2 x−2
e)
x 3−x
x 2− x
f)
x 2−4
x 2−4x4
Adding and subtracting algebraic fractions:
a) With the same denominator: You can add or subtract easily, simply add or subtract the numerators
and write the sum over the common denominator.
Examples: Calculate:
a)
3x4 x−4

x−3 x−3
b)
x −5 x −6
−
x 1 x1
2
2
b) With different denominators: Before you can add or subtract algebraic fractions with different
denominators, you must reduce to common denominator (calculate the LCM) and then add or
subtract numerators.
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Examples: Calculate:
a)
x
x−3

x−2 x1
b)
x−1
x

2
x −4 x2
Multiplying and dividing algebraic fractions:
P  x  R x P  x · R  x 
·
=
Q  x  S  x  Q x · S  x
P  x  R x  P  x · S  x 
:
=
Q  x  S  x Q x· R  x
Examples: Calculate:
a)
2x x5
·
x−3 x−1
b)
x 2 x −1
:
x1 x−2
Your
Turn
1. Calculate:
a)
x
x1
 2
x −1 x − x
b)
1
1
−
x −4 x−2
2
2
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
c)
x−1
x

x1 x−1
d)
x2
2x
−
x
x −2
e)
x x−4
·
x−2 x1
f)
x 2−3 2x2
·
x 1 x−3
g)
x−3 x 2−9
:
x4 x2
h)
x2
x
: 2
x−1 x −2x1
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Unit 3: Polynomials and Algebraic Fractions. Mathematics 4th E.S.O. Teacher: Miguel Ángel Hernández Lorenzo.
Keywords:
monomial = monomio
binomial = binomio
trinomial = trinomio
Polynomial = polinomio
variable = variable
constant = constante
the unknown = la incógnita
degree = grado
term = término
constant term = término independiente
Numerical value of a polynomial = valor numérico de un un polinomio
to plug in numbers for the variable = sustituir por números la variable
To evaluate … when x = … = calcular el valor … cuando x = …
like terms = términos semejantes
unlike terms = términos no semejantes
dividend = dividendo
divisor = divisor
quotient = cociente
remainder = resto
Ruffini's Rule = Regla de Ruffini
Remainder Theorem = Teorema del Resto
Factor Theorem = Teorema del Factor
Root or zeroes of a polynomial = raíces o ceros de un polinomio
to factorise = factorizar
common factor = factor común
common denominator = común denominador
to put fractions over a common denominator = escribir las facciones con
denominador común
to cross-multiply = multiplicar en cruz
Algebraic Fraction = Facción Algebraica
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