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Assembly of God Hebron Evening School
Mathematics Notes
More about Polynomials
Polynomials
A polynomial in one variable, say x, of degree n is an algebraic of the form
a n x n  a n 1 x n 1    a 2 x 2  a1 x  a0
Where n is a non-negative integer and the coefficients a n , a n1 ,, a1 , a0 are real
numbers.
Fundamental Operations
For two polynomials f(x) and g(x) :
Addition:
f(x) + g(x)
Subtraction: f(x) – g(x)
a polynomial
Multiplication: f(x) ‧ g(x)
f ( x) g ( x)
Division:
or
 NOT necessarily a polynomial
g ( x)
f ( x)
Equality of Polynomials
If two polynomials in x are equal for all values of x, then the two polynomials
are identical and, the coefficients of like powers of x in the two polynomials
must be equal.
e.g. If
Ax2 + Bx + C ≡ (2x-3)(x+5)
then Ax2 + Bx + C ≡ 2x2 + 7x – 15
A=2
B=7
C = -15
Theorems about Polynomials
A. Remainder Theorem
When a polynomial f(x) is divided by x – a , the remainder R is equal to
f(a).
Note: More generally, when a polynomial f(x) is divided by mx – n, the
n
remainder R is f( ).
m
Factor Theorem
If f(x) is a polynomial and f(a) = 0, then x – a is a factor of f(x)
Note:
(1) The converse of the Factor Theorem is also true, i.e. if x – a is a factor of the
polynomial f(x), then f(a) = 0.
n
(2) More generally, if mx – n is a factor of the polynomial f(x), then f ( )  0 .
m
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Assembly of God Hebron Evening School
Mathematics Notes
More about Polynomials
Applications of the Factor Theorem
A. Factorizing a Polynomial with Leading Coefficient Equal to 1
Procedures
Step 1
Find all the factors of the
constant term, say  1, 2, etc
Step 2
Find the values of f(1), f(2),
etc. one by one until one of them is
equal to 0.
Step 3
The factor is indentified, then
f(x) can be factorized by division.
e.g. Factorize
f(x) = x3 – x – 6
The factors are 1, 2, 3, 6.
We find f(2) = 0,
 x – 2 is a factor.
By division,
f(x) = (x – 2)(x2 + 2x +3)
B. Solving Cubic Equations
A cubic equation refers to an equation f(x) = 0 where f(x) is a polynomial of
degree 3. To solve f(x) = 0, we can make use of the Factor Theorem.
C. Finding H.C.F. and L.C.M of Polynomials
e.g. For two polynomials
x3 – 3x – 2 and x3 – 4x2 +4x:
x3 - 3x - 2 = (x – 2) (x + 1)2
x3 – 4x2 +4x = x(x – 2)2
H.C.F = x – 2
L.C.M = x (x – 2)2(x + 1)2
Harder Algebraic Fractions
An algebraic fraction is the one in which the denominator and the numerator
are non-zero polynomials. To simplify an algebraic fraction, we always need
to factorize the denominator and the numerator first.
e.g.
4 x 2  1 (2 x  1)( 2 x  1) 2 x  1


x(2 x  1)
x
2x 2  x
Fractional Equations
To solve fractional equations, we must reject values of the variable that would
make the denominator zero.
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Assembly of God Hebron Evening School
Mathematics Notes
More about Polynomials
e.g.For
1
x4

0
x( x  3) 3( x  3)
the restrictions are x  0 and x -3.
We have
3 + x (x + 4) = 0
x2 + 4x + 3 = 0
( x + 1)(x + 3)= 0
x = -1 or –3 (rejected)
x = -1 is the only solution of the equation.
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