MATH PROJECT WORKNAME - SHUBHANSHU BHARGAVACLASS -10SECTION - ASHIFT- 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 1÷x − 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- 2x−3 , ∫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 OFTHE 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 a>0 or a<0 .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 BETWEENZEROES OF A POLYNOMIAL For a quadratic polynomial – In general, if α and β are the zeroes of a quadratic polynomial p(x) = ax² + bx + c , a ≠ 0 , then we know that x − α and x− β are the factors of p(x) . Therefore ,• ax² + bx + c = k ( x − α) ( x − β ) ,• Where k is a constant = k[x² − (α + β)x +αβ]• = kx² − k( α + β ) x + k αβ• 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 ZEROAND COEFFICIENT OF A CUBIC POLYNOMIAL• In general, if α , β , Y are the zeroes of a cubic polynomial ax³ + bx² + cx + d , then∀ α+β+Y = − b÷a• = − ( Coefficient of x² ) ÷ coefficient of x³∀ αβ +βY +Yα =c ÷ a• = coefficient of x ÷ coefficient of x³∀ αβY = − 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 .