Polynomials

Name - Ankit GoelClass – X DRoll No. - 18

Submitted To: Ms. Neeru Dixit

Polynomials

PolynomialsPOLYNOMIAL – A polynomial in one variable X is an algebraic expression in X of the form

NOT A POLYNOMIAL – The expression like 1x 1,x+2 etc are not polynomials .

Degree Of PolynomialDegree of polynomial - The highest power of x in p(x) is called the degree of the polynomial p(x).

EXAMPLE –1. F(x) = 3x +½ is a polynomial in the variable

x of degree 1.2. g(y) = 2y² ⅜ y +7 is a polynomial in the

variable y of degree 2 .

Constant PolynomialCONSTANT POLYNOMIAL – A polynomial of degree zero is called a constant polynomial.

EXAMPLE - F(x) = 7 etc .1. It is also called zero polynomial.2. The degree of the zero polynomial is not defined

.

LINEAR POLYNOMIAL – A polynomial of degree 1 is called a linear polynomial .

EXAMPLE - 2x3 , 3x +5 etc .The most general form of a linear

polynomialis ax b , a 0 ,a & b are real.

Linear Polynomial

QUADRATIC POLYNOMIAL – A polynomial of degree 2 is called quadratic polynomial .

EXAMPLE – 2x² 3x ⅔ , y² 2 etc . More generally , any quadratic polynomial in x with real coefficient is of the form ax² + bx + c, where a, b, c are real numbers and a 0

Quadratic Polynomial

CUBIC POLYNOMIAL – A polynomial of degree 3 is called a cubic polynomial .

EXAMPLE = 2 x³ , x³, etc .The most general form of a cubic polynomial with coefficients as real numbers is ax³ bx² cx d , a ,b ,c ,d are reals .

Cubic Polynomial

Zero Of A PolynomialA real number k is said to a zero of a polynomial p(x), if said to be a zero of a polynomial p(x), if p(k) = 0 . For example, consider the polynomial p(x) = x³ 3x 4 . Then, p(1) = (1)² (3(1) 4 = 0 Also, p(4) = (4)² (3 4) 4 = 0 Here, 1 and 4 are called the zeroes of the quadratic polynomial x² 3x 4 .

We know that a real number k is a zero of the polynomial p(x) if p(K) = 0 . But to understand the importance of finding the zeroes of a polynomial, first we shall see the geometrical meaning of –1) Linear polynomial .2) Quadratic polynomial3) Cubic polynomial

Geometrical Meaning Of A Zeroes Of A Polynomial

Geometrical Meaning Of Linear Polynomial

For a linear polynomial ax b , a 0, the graph of y = ax b is a straight line . Which intersect the x axis and which intersect the x axis exactly one point ( b 2 , 0 ) . Therefore the linear polynomial ax b , a 0 has exactly one zero .

Geometrical Meaning Of A Quadratic Polynomial

For any quadratic polynomial ax² bx c, a 0, the graph of the corresponding equation y = ax² bx c has one of the two shapes either open upwards or open downward depending on whether a0 or a0 .these curves are called parabolas .

Geometrical Meaning Of Cubic Polynomial

The zeroes of a cubic polynomial p(x) are the x coordinates of the points where the graph of y = p(x) intersect the x – axis . Also , there are at most 3 zeroes for the cubic polynomials . In fact, any polynomial of degree 3 can have at most three zeroes .

Relationship Between Zeroes Of A Polynomial

For a quadratic polynomial – In general, if and are the zeroes of a quadratic polynomial p(x) = ax² bx c , a 0 , then we know that x and x are the factors of p(x) . Therefore ,

ax² bx c = k ( x ) ( x ) ,Where k is a constant = k[x² ( )x ]= kx² k( ) x k Comparing the coefficients of x² , x and constant term on both the sides .Therefore , sum of zeroes = b a= (coefficients of x) coefficient of x² Product of zeroes = c a = constant term coefficient of x²

Relationship Between Zero And Coefficient Of Cubic Polynomial

In general, if , , Y are the zeroes of a cubic polynomial ax³ bx² cx d , then 1. Y = b a = ( Coefficient of x² ) coefficient of x³ 2. Y Y =c a= coefficient of x coefficient of x³3. Y = d a= constant term coefficient of x³

Division Algorithm Of Polynomials

If p(x) and g(x) are any two polynomials with g(x) 0, then we can find polynomials q(x) and r(x) such that – p(x) = q(x) g(x) r(x)Where r(x) = 0 or degree of r(x) degree of g(x) .This result is taken as division algorithm for polynomials .

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