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Presented By;-NAME – Ashish Pradhan , Durgesh KumarCLASS- X – ‘A’ROLL NO-27 , 26
A presentation on
INTRODUCTIONGEOMETRICAL MEANING OF ZEROES OF THE POLYNOMIAL
RELATION BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
DIVISION ALGORITHM FOR POLYNOMIAL
Polynomials are algebraic expressions that include real numbers and variables. The power of the variables should always be a whole number. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.Polynomials contain more than one term. Polynomials are the sums of monomials. A monomial has one term: 5y or -8x2 or 3. A binomial has two terms: -3x2 2, or 9y - 2y2
A trinomial has 3 terms: -3x2 2 3x, or 9y - 2y2 y The degree of the term is the exponent of the variable: 3x2 has a
degree of 2.When the variable does not have an exponent - always understand that there's a '1' e.g., 1x
Example:x2 - 7x - 6 (Each part is a term and x2 is referred to as the leading term)
WHAT IS A POLYNOMIAL
A polynomial is an expression made with constants, variables and exponents, which are combined using addition, substraction and mutiplication but not division.
The exponents can only be 0,1,2,3…. etc.
A polynomial cannot have infinite number of terms.
Let “x” be a variable and “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.
NOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.
DIFFERENT TYPES OF
POLYNOMIALS
ON THE BASIS OF NUMBER OF TERMS—
o MONOMIAL – POLYNOMIALS HAVING ONLY ONE TERM. E.G. 4X, 8Y
o BINOMIAL – POLYNOMIALS HAVING TWO TERMS. E.G. 2X + 6, 25Y – 25
o TRINOMIAL – POLYNOMIALS HAVING THREE TERMS. E.G. 2X - X³ +25, X³ + 5X² -8
The degree is the term with the greatest exponent
Recall that for y2, y is the base and 2 is the exponent
For example: p(x) = 10x4 + ½ is a polynomial in the variable x of degree 4.
p(x) = 8x3 + 7 is a polynomial in the variable x of degree 3.
p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in the variable x of degree 3.
p(x) = 8u5 + u2 – 3/4 is a polynomial in the variable x of degree 5.
DEGREE
i) Constant polynomial – polnomials having degree 0.
e.g. 32, -5
ii) Linear polynomial – polynomials having degree 1.
e.g. x+5, 6x-3
ii) quadratic polynomial – polynomials having degree 2.
e.g. 2x² + 3x -8
iii) Cubic polynomial – polynomials having degree 3.
e.g. 6x³ + 7x² -x-6
v) bi-quadratic polynomial- polynomials having degree 4.
e.g. 2x4 + x³ - 8x² +5x -8
More information of degree
ZEROES OF A POLYNOMIAL
A real number α is a zero of a
polynomial f(x), if f(α) = 0.
e.g. f(x) = x³ - 6x² +11x -6
f(2) = 2³ -6 X 2² +11 X 2 – 6
= 0 .Hence 2 is a zero
of f(x).
The number of zeroes of the
polynomial is the degree of the polynomial. Therefore a quadratic
polynomial has 2 zeroes and cubic
3 zeroes.
For example: f(x) = 7, g(x) = -3/2, h(x) = 2are constant polynomials. The degree of constant polynomials is
ZERO.
For example: p(x) = 4x – 3, p(y) = 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 polynomial is always in the form:-
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) = 5x3 – 2x2 + 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.
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 p(x) is a polynomial and “y” is any real no. then real no. obtained by replacing “x” by “y”in p(x) is called the value of p(x) at x = y and is denoted by “p(y)”.
For example:-Value of p(x) at x = 1 p(x) = 2x2 – 3x – 2 p(1) = 2(1)2 – 3 x 1 – 2 = 2 – 3 – 2 = -3
For example:-Zero of the polynomial f(x) = x2 + 7x +12 f(x) = 0 x2 + 7x + 12 = 0 (x + 4) (x + 3) = 0 x + 4 = 0 or, x + 3 = 0x = -4 , -3
VALUE OF POLYNOMIAL
ZERO OF A POLYNOMIAL
QUADRATIC
☻ A + B = - Coefficient of x
Coefficient of x2
= - ba
☻ AB = Constant term Coefficient of x2
= ca
Note:- “A” and “B” are the zeroes.
RELATIONSHIP BETWEEN THE ZEROES AND COEFFICIENTS OF A
QUADRATIC POLYNOMIAL
An nth degree polynomial can have at most “n” real zeroes.
Graphs of the polynomialsNumber of real zeroes of a polynomial is less than or equal to degree of the polynomial.
GENERAL SHAPES OF POLYNOMIAL
FUNCTIONS f(x) = x + 2
LINEAR FUNCTION
DEGREE =1
MAX. ZEROES = 1
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QUADRATIC FUNCTION
DEGREE = 2
MAX. ZEROES = 2
Relationship between the zeroes and coefficients of a cubic polynomial
• Let α, β and γ be the zeroes of the polynomial ax³ + bx² + cx + d
• Then, sum of zeroes(α+β+γ) = -b = -(coefficient of x²)
a coefficient of x³
αβ + βγ + αγ = c = coefficient of x
a coefficient of x³
Product of zeroes (αβγ) = -d = -(constant term)
a coefficient of x³
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CUBIC FUNCTION
DEGREE = 3
MAX. ZEROES = 3
ON VERYFYING THE
RELATIONSHIP BETWEEN
THE ZEROES AND
COEFFICIENTS
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.
RELATIONSHIPS.
DIVISION ALGORITHM FOR POLYMIALS
If p(x) and g(x) are any two polynomials with g(x) ≠ 0,then we can always find polynomials q(x), and r(x) such that :
P(x) = q(x) g(x) + r(x),Where r(x) = 0 or degree r(x) < degree g(x)
QUESTIONS BASED ON POLYNOMIALS
I) Find the zeroes of the polynomial x² + 7x + 12and verify the relation between the zeroes and its coefficients.
f(x) = x² + 7x + 12
= x² + 4x + 3x + 12
=x(x +4) + 3(x + 4)
=(x + 4)(x + 3)
Therefore,zeroes of f(x) =x + 4 = 0, x +3 = 0 [ f(x) = 0]
x = -4, x = -3
Hence zeroes of f(x) are α = -4 and β = -3.
Sum of zeroes = α + β = -4 -3 = -7 -(coefficient of x) = -7 coefficient of x²Hence, sum of zeroes = -(coefficient of x) coefficient of x²Product of zeroes = αβ = (-4)(-3) = 12Constant term = 12Coefficient of x²Hence, product of zeroes = constant term coefficient of x²
2) Find a quadratic polynomial whose zeroes are 4, 1.
sum of zeroes,α + β = 4 +1 = 5 = -b/a
product of zeroes, αβ = 4 x 1 = 4 = c/a
therefore, a = 1, b = -4, c =1
as, polynomial = ax² + bx +c
= 1(x)² + { -4(x)} + 1
= x² - 4x + 1
2) Find a quadratic polynomial whose zeroes are 4, 1.
sum of zeroes,α + β = 4 +1 = 5 = -b/a
product of zeroes, αβ = 4 x 1 = 4 = c/a
therefore, a = 1, b = -4, c =1
THE END