polynomials algebra
TRANSCRIPT
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions.
Introduction :
A polynomial is an expression of finite length constructed from
variables and constants, using only the operations of addition,
subtraction, multiplication, and non-negative, whole-number exponents.
Polynomials appear in a wide variety.
Let x be a variable n, be a positive integer and as, a1,a2,….an be constants (real nos.)
Then, f(x) = anxn+ an-1xn-1+….+a1x+xo
anxn,an-1xn-1,….a1x and ao are known as the terms of the polynomial.
an,an-1,an-2,….a1 and ao are their coefficients.
For example:
• p(x) = 3x – 2 is a polynomial in variable x.
• q(x) = 3y2 – 2y + 4 is a polynomial in variable y.
• f(u) = 1/2u3 – 3u2 + 2u – 4 is a polynomial in variable u.
NOTENOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.
The exponent of the highest degree term in a polynomial is known as its degree.
For example:
f(x) = 3x + ½ is a polynomial in the variable x of degree 1.
g(y) = 2y2 – 3/2y + 7 is a polynomial in the variable y of degree 2.
p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in the variable x of degree 3.
q(u) = 9u5 – 2/3u4 + u2 – ½ is a polynomial in the variable u of degree 5.
Polynomials in one variable
A polynomial is a monomial or a sum of monomials.
Each monomial in a polynomial is a term of the polynomial.
The number factor of a term is called the coefficient.
The coefficient of the first term in a polynomial is the lead coefficient.
A polynomial with two terms is called a binomial. A polynomial with three term is called a trinomial.
Polynomials in one variable
The degree of a polynomial in one variable is the largest exponent of that variable.
1425 2 −+ xx
14 +x
A constant has no variable. It is a 0 degree polynomial.2This is a 1st degree polynomial. 1st degree polynomials are linear.
This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.
183 3 −x This is a 3rd degree polynomial. 3rd degree polynomials are cubic.
For example:f(x) = 7, g(x) = -3/2, h(x) = 2
are constant polynomials.The degree of constant polynomials is not defined.
For example: p(x) = 4x – 3, q(x) = 3y are linear polynomials.Any linear polynomial is in the form ax + b, where a, b are real nos. and a ≠ 0.
It may be a monomial or a binomial. F(x) = 2x – 3 is binomial whereas g (x) = 7x is monomial.
A polynomial of degree two is called a quadratic polynomial.
f(x) = √3x2 – 4/3x + ½, q(w) = 2/3w2 + 4 are quadratic polynomials with real coefficients.
Any quadratic is always in the form f(x) = ax2 + bx +c where a,b,c are real nos. and a ≠ 0.
A polynomial of degree three is called a cubic polynomial.
f(x) = 9/5x3 – 2x2 + 7/3x _1/5 is a cubic polynomial in variable
x.
Any cubic polynomial is always in the form f(x = ax3 + bx2 +cx + d where a,b,c,d are
real nos.
Examples
Polynomials Degree Classify by degree
Classify by no. of terms.
5 0 Constant Monomial
2x - 4 1 Linear Binomial
3x2 + x 2 Quadratic Binomial
x3 - 4x2 + 1 3 Cubic Trinomial
A real no. x is a zero of the polynomial f(x), is f(x) = 0 Finding a zero of the polynomial means solving polynomial equation f(x) = 0.
If f(x) is a polynomial and y is any real no. then real no. obtained by replacing x by y in f(x) is cal led the value of f(x) at x = y and is denoted by f(x).
Value of f(x) at x = 1 f(x) = 2x 2 – 3x – 2 f(1) = 2(1) 2 – 3 x 1 – 2 = 2 – 3 – 2 = -3
Zero of the polynomial f(x) = x 2 + 7x +12 f(x) = 0 x2 + 7x + 12 = 0 (x + 4) (x + 3) = 0 x + 4 = 0 or, x + 3 = 0x = -4 , -3
• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In this case, a is also called a root of the equation p(x)=0.
• Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.
Number of TurnsAnother fact, that you’ll justify in Calculus I is that the graph of a polynomial of degree n can have at most n – 1 turns in it (where we switch from increasing to decreasing or back). The number of turns may be smaller than n – 1, but only by an even number.
Degree 1 0 turns Degree 2 1 turn
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QUADRATIC FUNCTION
DEGREE = 2
MAX. ZEROES = 2
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CUBIC FUNCTION
DEGREE = 3
MAX. ZEROES = 3
POLYNOMIAL FUNCTIONSGENERAL SHAPES OF POLYNOMIAL FUNCTIONSf(x) = x4 + 4x3 – 2x – 1 QuaDrticFunctionDegree = 4Max. Zeros: 4
POLYNOMIAL FUNCTIONSGENERAL SHAPES OF POLYNOMIAL FUNCTIONSf(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 QuinticFunctionDegree = 5Max. Zeros: 5
Typical graph of a linear polynomial : y= 2x + 3
DPlotTrialVersionhttp://www.dplot.com
Polynomials1.grf
y=2x+3
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
5
6
7
8
9
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
5
6
7
8
9
Y-axis where all the x coordinates are 0
X-axis where all the y coordinates are 0
(-1.5,0)i.e.
x=-1.5; y=0
(0,3)i.e. x=0; y=3
0,0
Quadratic polynomial: ax2+bx+c
Y=x2-3x-4Y=x2-2x-3
These curves are called parabolas
Intercept in plain english means “cut”
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-8
-4
0
4
8
12
16
20
24
28
32
36
40
44
48
52
No intercept on x-axis
Types of quadratic polynomials
Graph a) has two real Zeros. It has two x-intercepts.Graph b) has no real zeros. It has no x-intercepts. Both zeros are complex.
Graph c) the intercept shows only one zero. Though it is a polynomial of two degree it has only one zero!
How many
Zeros? in this
polynomial
No Zero in this case it does not
intercept the x axis
Two zeros as the curve
intercepts at two points on the x =axis
The maximum zeros the second degree polynomial has is TWO
Cubic Polynomial: p(x)=x3-4x
DPlotTrialVersionhttp://www.dplot.com
Plot2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Points where the curve interceptsThe x – axis:
The zeros are: -2.0,2
Thus the third degree polynomial has three zeros( roots)
QUADRATICQUADRATIC
☻ α + β = - coefficient of x
Coefficient of x2
= - ba
☻ αβ = constant termCoefficient of x2
= ca
CUBICCUBIC
α + β + γ = -Coefficient of x2 = -bCoefficient of x3 a
αβ + βγ + γα = Coefficient of x = cCoefficient of x3 a
αβγ = - Constant term = dCoefficient of x3 a
ON VERYFYI
NG THE
RELATIONSHIP B
ETWEEN
THE ZEROES A
ND
COEFFICIEN
TS
ON FINDING THE
VALUES OF EXPRESSIONS
INVOLVING ZEROES OF
QUADRATIC POLYNOMIAL
ON FINDING AN
UNKNOWN WHEN A
RELATION BETWEEEN
ZEROES AND COEFFICIENTS
ARE GIVEN.
OF ITS A QUADRATIC
POLYNOMIAL WHEN
THE SUM AND
PRODUCT OF ITS
ZEROES ARE GIVEN.
If f(x) and g(x) are any two polynomials with g(x) ≠ 0,then we can always f ind polynomials q(x), and r(x) such that :
F(x) = q(x) g(x) + F(x) = q(x) g(x) + r(x),r(x) ,
Where r(x) = 0 or degree r(x) < degree g(x)
ON VERYFYING THE DIVISION ALGORITHM FOR POLYNOMIALS.
ON FINDING THE QUOTIENT AND REMAINDER USING DIVISION ALGORITHM.
ON CHECKING WHETHER A GIVEN POLYNOMIAL IS A FACTOR OF THE OTHER POLYNIMIAL BY APPLYING THEDIVISION ALGORITHM
ON FINDING THE REMAINING ZEROES OF A POLYNOMIAL WHEN SOME OF ITS ZEROES ARE GIVEN.
F(x) = P(x) G(x) + R(x)
D. Theorm Remainder
If polynomials F(x) = P(x) H(x) + R and P(x) = (x-h), so
we will get : F(x) = (x-h). H(x) + RTHEORM : if the polynomials F(x) is divided by
(x-h), the remainder is F(h)Prove: F(x) = (x-h) . H(x) + R
F(h) = (h-h) . H(x) + R = 0 + R
F(h) = R
G. Factor Theorm
Theorm :If F(x) a polynomials then F(h) = 0
If and only if (x-h) is factor of F(x)
I. If F(h) =0 then (x-h) is factor of F(x)Prove:
F(x) = P(x) H(x) + R= (x-h) H(x) + R
F(h) = 0 = (x-h) H(x) + 0= (x-h) H(x)
Proved (x-h) is factor F(x)
H. Determined factor of Polynomials
Algorithm :1. If the sum coefficient xodd = the sum of
coefficient xeven so x = 1 is the factor of polynomials.
2. If the sum of coefficient xodd = sum of coefficient xeven so x=-1 is the factor of
polynomials.3. If the 1st and 2nd step not fulfill look at factor
of constant (by chance). Until find R of it is 0.
Algebraic Identities
Some common identities used to factorize polynomials
(x+a)(x+b)=x2+(a+b)x+ab(a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b)