MATH PROJECT WORK

NAME - SHUBHANSHU BHARGAVA

CLASS -10

SECTION - A

SHIFT- I SHIFT

POLYNOMIALS

• POLYNOMIAL – 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 POLYNOMIAL

• Degree 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 .

TYPES OF POLYNOMIALS

• Types of polynomials are –

• 1] Constant polynomial

• 2] Linear polynomial

• 3] Quadratic polynomial

• 4] Cubic polynomial

• 5] Bi-quadratic polynomial

CONSTANT POLYNOMIAL

• CONSTANT POLYNOMIAL – A polynomial of degree zero is called a constant polynomial.

• EXAMPLE - F(x) = 7 etc .

• It is also called zero polynomial.

• The degree of the zero polynomial is not defined .

LINEAR POLYNOMIAL

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

• EXAMPLE- 2x3 , 3x +5 etc .

• The most general form of a linear polynomial is ax b , a 0 ,a & b are real.

QUADRATIC 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

CUBIC POLYNOMIALS

• 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 .

BI QUADRATIC POLYNMIAL

• BI – QUADRATIC POLYNOMIAL – A fourth degree polynomial is called a biquadratic polynomial .

VALUE OF POLYNOMIAL

• If p(x) is a polynomial in x, and if k is any real constant, then the real number obtained by replacing x by k in p(x), is called the value of p(x) at k, and is denoted by p(k) . For example , consider the polynomial p(x) = x² 3x 4 . Then, putting x= 2 in the polynomial , we get p(2) = 2² 3 2 4 = 4 . The value 6 obtained by replacing x by 2 in x² 3x 4 at x = 2 . Similarly , p(0) is the value of p(x) at x = 0 , which is 4 .

ZERO OF A POLYNOMIAL

• A 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 .

HOW TO FIND THE ZERO OF A LINEAR POLYNOMIAL

• In general, if k is a zero of p(x) = ax b, then p(k) = ak b = 0, k = b a . So, the zero of a linear polynomial ax b is b a = ( constant term ) coefficient of x . Thus, the zero of a linear polynomial is related to its coefficients .

GEOMETRICAL MEANING OF THE ZEROES OF A POLYNOMIAL

• 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 polynomial• 3) Cubic 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 .

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 A CUBIC

POLYNOMIAL• In general, if , , Y are the zeroes of a

cubic polynomial ax³ bx² cx d , then Y = ba • = ( Coefficient of x² ) coefficient of x³ Y Y =c a• = coefficient of x coefficient of x³ Y = d a• = constant term coefficient of x³

DIVISION ALGORITHEM FOR 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 .

THE END